Zsolt Szilvágyi - Széchenyi István University · Zsolt Szilvágyi Dynamic Soil Properties of Danube Sands ii Acknowledgments First, I would like to extend my appreciation and gratitude

Zsolt Szilvágyi

Dynamic Soil Properties of Danube Sands

PhD Dissertation

Supervisor:

Professor Dr. Richard P. Ray PhD

Department of Structural and Geotechnical Engineering

Faculty of Architecture, Civil Engineering and Transport Sciences

Széchenyi István University

Modeling and Development of Infrastructural Systems

Multidisciplinary Doctoral School of Engineering Sciences

Széchenyi István University

Győr, Hungary

2018

Zsolt Szilvágyi Dynamic Soil Properties of Danube Sands

ii

Acknowledgments

First, I would like to extend my appreciation and gratitude to my supervisor, Prof. Richard P.

Ray. He introduced me to the challenging topics of soil dynamics and laboratory testing and

guided me through this research with valuable oversight, while allowing the very necessary

independent research as well.

I would also like to thank Dr. Róbert Szepesházi, for bringing me into his Geotechnical

Workshop in Győr. Our collective work induced great professional challenges and has resulted

in many fruitful discussions and exciting experiences. I would like to express my gratitude to

my colleagues in Győr at the Department of Structural and Geotechnical Engineering, espe-

cially Ákos Wolf, Dr. Edina Koch, Péter Hudacsek and Dr. Orsolya Kegyes-Brassai for all their

help and collaboration.

I would also like to thank my fellow PhD student, Jakub Panuska, from the Department

of Geotechnics at the Slovak University of Technology, for his always-positive attitude and for

the enormous work that he contributed to our mutual research papers.

Finally, my sincere thanks to my family for their love and support; and for encouraging

me to challenge myself repeatedly.


iii

Summary

This PhD dissertation concerns with the dynamic behavior of Danube Sands. Recently in Hun-

gary the adaptation of Eurocode 8 standard has created a greater need to define dynamic behav-

ior of local soils such as Danube Sands. Seismicity in Hungary can be considered moderate,

however expected earthquake loading could cause serious damage in structures and infrastruc-

ture, therefore engineers need to consider it in design. Due to the fact, that the previous Hun-

garian design standards did not consider earthquake loading, there has not been any research

focusing on soil dynamics in Hungary, and their lack hinders economic design and construction

of engineering structures, and the precise specification of local earthquake risk.

In order to obtain information about the response of Danube Sands to dynamic and cyclic

loading, state-of-the art laboratory measurement methods have been applied and the most im-

portant properties needed to describe dynamic behavior were measured. The laboratory inves-

tigations included the use of the Resonant Column test method and the Torsional Simple Shear

test method. Both methods are used widely around the world for dynamic laboratory testing,

however they have not been used before in Hungary. The device used for testing can be con-

sidered unique worldwide, since it is capable of performing both tests on a single hollow cylin-

der soil specimen, while commonly used devices are usually capable of performing only one

type of test. The most important feature of the behavior of soil subjected to dynamic loading is

that deformation response is depending on the induced level of deformations. In other words

stiffness is depending on strain and their relationship is highly nonlinear. The biggest advantage

of the combined device used in this study, is the capability of measuring soil response over a

very wide range of strains, which makes it possible to define soil behavior more precisely.

The specific scope of the study was to measure the very small strain stiffness, the stiffness

degradation and the damping characteristics of Danube Sands and to obtain correlations which

describe how these parameters depend on state variables. This was achieved by the development

of measurement data interpretation procedures, comparison of measurement results to existing

correlations and development of correlations to better describe these local soils. Since the meas-

urement techniques are quite complex, data interpretation is an important task in the procedure.

Interpretation includes the task of choosing material model parameters which requires the so-

lution of an inverse problem. To solve the problem, a thorough literature review on the model-

ing of small strain stiffness had to be performed and a procedure was developed which allows

obtaining parameters of a nonlinear material model, the Ramberg-Osgood model, based on the

laboratory measurement results.


iv

Összefoglalás

Disszertációm Dunai homokok dinamikus viselkedésével foglalkozik. Hazánkban az Eurocode

8 szabvány bevezetése miatt jelentős igény merült fel a helyi talajok dinamikus viselkedésének

leírására. A földrengésveszély Magyarországon mérsékeltnek tekinthető, ugyanakkor a várható

földrengések komoly károkat okozhatnak szerkezeteinkben és az infrastruktúra hálózatban,

ezért a mérnöki tervezés során figyelembe kell venni a földrengésterhet. Mivel a korábbi ma-

gyar szabványok nem számoltak ezzel a teherrel, a talajdinamika témaköréhez kapcsolódó ha-

zai kutatás eddig még nem volt. A témára vonatkozó kutatások hiánya nem csak a szerkezetek

gazdaságos tervezését és kivitelezését gátolja, de nehezíti a földrengésveszély helyi értékének

pontos meghatározását is.

Kutatásomban a legkorszerűbb laboratóriumi mérési módszereket alkalmaztam tipikus

Dunai homokok dinamikus és ciklikus terhelésekre adott válaszának jellemzéséhez és megmér-

tem a dinamikus viselkedés leírásához szükséges legfontosabb talajjellemzőket. Laboratóriumi

méréseim Rezonanciás vizsgálatból és Torziós Nyírásvizsgálatból álltak. Mindkét mérési mód-

szer fejlett dinamikai laboratóriumi mérési módszernek számít az egész világon, ugyanakkor

Magyarországon még nem használták egyiket sem. A kutatáshoz használt és továbbfejlesztett

mérőberendezés pedig világszinten is egyedinek számít, mivel a két mérési módszert egyetlen

üreges hengeres talajmintán képes egymás után alkalmazni, míg az elterjedt berendezések

szinte mind csak az egyik mérést képesek elvégezni. A dinamikus terhelésnek kitett talajok

viselkedésének legfontosabb jellemzője, hogy a terhelés okozta elmozdulások mértéke függ az

elmozdulások nagyságától. Másképp megfogalmazva, a talaj merevsége függ a fajlagos alak-

változásoktól és a kettejük közötti kapcsolat erősen nemlineáris. A kutatáshoz használt és to-

vábbfejlesztett berendezés legnagyobb előnye, hogy a talaj dinamikus teherre adott válaszát a

fajlagos alakváltozások mértékének széles tartományában képes megmérni és ez által ponto-

sabban le lehet írni a talaj viselkedését.

Kutatásom legfőbb célja volt tipikus Dunai homokok nagyon kis alakváltozások tartomá-

nyában érvényes nyírási modulusának mérése, továbbá a modulus növekvő alakváltozások mel-

letti leromlási és a csillapítási jellemzőinek mérése, valamint olyan korrelációk felállítása, me-

lyek leírják, hogyan függnek ezek a mennyiségek a befolyásoló állapotjellemzőktől. Ehhez

szükséges volt a mérési eredmények interpretációjának továbbfejlesztése, valamint a mérési

eredményeket össze kellett vetni meglévő, szakirodalomban található korrelációkkal és felállí-

tani új korrelációkat, amelyek jobban leírják ilyen talajok viselkedését. Mivel a mérési módsze-

rek meglehetősen bonyolultak, az eredmények interpretációja fontos feladata a folyamatnak.

Ebbe bele tartozik a választott anyagmodell paramétereinek megválasztása is, amely egy inverz


v

feladat megoldását igényli. A megoldáshoz irodalomkutatást végeztem a kis alakváltozások

esetén érvényes, nemlineáris modulus modellezésére vonatkozóan és kidolgoztam egy eljárást,

amellyel a nemlineáris Ramberg-Osgood anyagmodell paraméterei pontosan meghatározhatók

a laboratóriumi mérési eredmények alapján.


vi

Contents:

1 Introduction ........................................................................................................................... 1

1.1 Background ........................................................................................................ 1

1.2 Scope of study and organization of thesis .......................................................... 3

1.3 Research methodology ....................................................................................... 4

2 Dynamic behavior of soils ..................................................................................................... 5

2.1 Small strain nonlinearity .................................................................................... 7

2.1.1 Modeling small strain nonlinearity .......................................................... 11

2.1.2 Equivalent linear model............................................................................ 13

2.1.3 Modeling irreversible small strain behavior ............................................. 17

2.1.4 Modified Ramberg-Osgood model ........................................................... 18

2.2 Measurement of dynamic soil properties ......................................................... 22

2.2.1 In-situ tests ............................................................................................... 23

2.2.2 Laboratory tests ........................................................................................ 26

2.3 Determination of dynamic soil parameters based on correlations ................... 28

2.3.1 Small strain stiffness ................................................................................ 29

2.3.2 Modulus reduction .................................................................................... 35

2.3.3 Damping ................................................................................................... 42

3 RC-TOSS testing ................................................................................................................. 44

3.1 Basic testing concepts of RC and TOSS .......................................................... 44

3.2 Device ............................................................................................................... 46

3.3 Sample preparation ........................................................................................... 49

3.4 Interpretation of measurement results .............................................................. 50

3.5 Application of measurement results in numerical modeling ............................ 55

4 Testing program and results ................................................................................................ 65

4.1 Materials investigated and test matrix .............................................................. 65

4.2 Measured Gmax compared to correlations from literature and BE .................... 69

4.3 Modulus reduction curves ................................................................................ 82

4.4 Damping curves ................................................................................................ 97

5 Thesis statements ............................................................................................................... 110

5.1 Thesis #1 ........................................................................................................ 111

5.2 Thesis #2 ........................................................................................................ 111

5.3 Thesis #3 ........................................................................................................ 112

5.4 Thesis #4 ........................................................................................................ 112


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5.5 Thesis #5 ........................................................................................................ 113

6 Summary and future research ............................................................................................ 113

References .............................................................................................................................. 115

List of figures ......................................................................................................................... 124

List of symbols ....................................................................................................................... 128

Annex A – Device assembly and sample preparation for RC-TOSS testing ..............................a

Annex B – Developed Visual Basic codes .................................................................................. s

Annex C – RC-TOSS measurement results .............................................................................. w


1

1 Introduction

"The finding that at small strains the ground exhibits high stiffness and, frequently, nonlinear

properties, has far-reaching practical and fundamental consequences. ...nonlinearity of stress-strain

response has very significant effects on soil-structure interaction, stress distributions in the soil mass,

and displacement profiles around loaded areas and excavations. ...There is enormous scope for the

development of improved instruments and techniques and for the experimental study of stiffness prop-

erties of soils and weak rocks at small strains." (Burland, 1989)

“The limit of classical laboratory testing coincides … with characteristic shear strains that can

be measured near geotechnical structures. However, the soil stiffness that should be used in the analysis

of geotechnical structures is not the one that relates to these final strains. Instead, very small-strain soil

stiffness and its non-linear dependency on strain amplitude should be properly taken into account in all

analysis that strive for reliable predictions of displacements.” (Benz, 2006)

1.1 Background

The adaptation of Eurocode 8 for seismic design and the continuous development of the rail

system in Hungary have created a greater need to define dynamic behavior of local soil groups.

Seismic design includes geotechnical and structural engineering tasks e.g. design of founda-

tions, retaining structures, buildings and other engineering structures for earthquake loads; as

well as earthquake engineering tasks e.g. site response evaluation, seismic risk assessment etc.

Most of these tasks require knowledge about dynamic behavior of soils to some extent. Seismic

design and earthquake engineering are challenging tasks for the designer. Earthquake loading

acts over a very short period of time compared to other loads and its rare occurrence makes it

difficult to gather useful design data.

This is especially true in Hungary where seismic risk can be considered moderate, but it

is high enough to impose a risk, which engineers must address. Meeting the especially strict

design criteria concerning vibrations and settlements set for high-speed rails also require com-

prehensive knowledge about geotechnical dynamic modeling. This topic will gain more atten-

tion in the near future since the development of the Hungarian high-speed rail system is essen-

tial and will be required from us by the European Union. Due to these tendencies, several topics

of seismic design have been studied in Hungary recently, such as in-situ geophysical measure-

ment methods (Törös, 2006), (Kegyes-Brassai, Ray, & Tildy, 2015); seismic design of struc-


2

tures (Zsarnóczay, 2014), (Simon & Vigh, 2017), (Ther & Kollár, 2017); and liquefaction as-

sessment (Bán, Katona, & Mahler, 2016); however, measurement and modeling of local soils

under dynamic loading has never been studied yet.

Recently, the importance of numerical modeling in geotechnical engineering has in-

creased greatly in research studies and everyday practice as well. As a result, geotechnical ma-

terial models have developed tremendously to meet the increasing demands of numerical mod-

eling. On the one hand, this is a result of the rapid and continuous progress of computer tech-

nology; on the other hand, the many advantages of achieving more accurate and economical

solutions help these approaches become more widespread. Generally, models developed for

scientific calculations tend to get implemented into commercial software faster and faster.

However, another tendency can be observed in Hungary, and perhaps elsewhere, that it

is difficult to convince clients to spend on elaborate laboratory testing programs even in larger

projects. This may be due to a lack of appreciation of the complexities of geotechnical modeling

by engineers from related fields; they don’t understand the difficulties or the benefits of a more

sophisticated model. It is the author’s hope that this study will help practicing engineers under-

stand, how material models can be used to describe the dynamic behavior of soils and how

small strain stiffness can be assessed in the laboratory. Besides, since there is often no room in

the project for complex and time-consuming laboratory testing, the study will also provide cor-

relations which can be used to estimate dynamic material parameters of soils similar to those

tested and analyzed in this study.

In the past 7-8 years, a research team has been formed at the Department of Structural

and Geotechnical Engineering at Széchenyi István University under the supervision of Prof.

Richard P. Ray. My research focuses on laboratory testing and modeling connected to small

strain stiffness of a typical local granular soil, while other team members consider related top-

ics, such as: earthquake hazard analysis and seismic risk assessment of existing buildings

(Kegyes-Brassai, 2016), pile foundations under seismic loading (Wolf & Ray, 2017) and finite

element modeling of dynamic loading of trains (Koch & Hudacsek, 2017).

For the research presented in my Dissertation, the continuous development of the Ge-

otechnical Laboratory was essential, especially the improvement of the combined Resonant

Column-Torsional Simple Shear device which was built by Prof. Ray at the University of Mich-

igan (Ray R. P., 1983) and rebuilt, recalibrated and further developed by the author at the Ge-

otechnical Laboratory of Széchenyi István University with the supervision of Prof. Ray.


3

1.2 Scope of study and organization of thesis

The scope of the study concerns laboratory testing and modeling of selected granular soils from

Hungary. Danube sands compose an important segment of local soils and present the geotech-

nical engineer with several design challenges. These soils are present at river crossings and

major development parcels throughout Hungary. Although their engineering behavior can vary

over a wide range, they retain some common characteristics that will help the engineer make

decisions about geotechnical and structural designs. Field and laboratory measurements are

usually complex and time consuming. In practice, the designer often has to use existing corre-

lations derived from testing of various soils. Therefore, my research aims to answer the follow-

ing questions:

Is there a material model available, which may be used to model the dynamic behavior of

Danube sands?

Can existing correlations be used to obtain reliable material parameters for the model?

How can the dynamic properties of Danube sands be quantified by laboratory measure-

ments?

To answer these questions, a logical order of general theory, literature review, laboratory

testing methods and equipment, data interpretation and finally test results and discussion will

be presented.

In Chapter 2, first the dynamic behavior of soils is discussed from a theoretical point-of-

view. Then applicable soil models are presented and the measurement methods for model pa-

rameters are detailed together with a review of literature on correlations. The aim of this chapter

is to introduce the reader to the modeling of small strain nonlinearity for soils as well as draw

attention to appropriate correlations for dynamic soil properties of sands and provide insight

about likely errors involved from such estimations.

Chapter 3 first presents the basic testing methodology used for the Resonant Column and

Torsional Shear testing. Next, the laboratory testing device used in this study is described and

testing procedure is presented. Then, since data interpretation is very important in this challeng-

ing field of measurements, it is shown how measured data can be turned into material parame-

ters. Finally, as the last step of interpretation, the inverse problem of obtaining model parame-

ters for a numerical model is solved and presented within the finite element modeling of a Tor-

sional Shear Test.

Chapter 4 presents the performed comprehensive laboratory testing program, starting

with the description and index parameters of the selected typical Danube sands. Then main


4

results of the measurements are summarized and compared to correlations from literature. Three

main branches of results are presented separately: small strain stiffness, modulus degradation

and damping of the soils are described. The summary of a comprehensive study is also shown,

which concerned the comparison of the test results to an independent set of laboratory meas-

urements made with Bender Elements at the Slovak University of Technology in Bratislava by

an independent researcher.

Summary of findings and thesis statements are given in Chapter 5.

Device assembly and sample preparation is further detailed in Annex A. Developed VBA

codes for data interpretation are given in Annex B. All laboratory test results are documented

in Annex C.

1.3 Research methodology

In order to provide background to the topic of the research and to gather existing correlations

for the dynamic properties of soils, a comprehensive literature review has been performed. The

review focused on two main topics: the theoretical description of modeling the dynamic behav-

ior of soils, and correlations for obtaining material model parameters for sands.

The research consisted of an extensive laboratory measurement program; two laboratory

measurement methods (Resonant Column Test and Torsional Simple Shear Test) have been

used to obtain information about the dynamic behavior of nine selected Danube sands. Both

methods are considered state-of-the-art in the field of geotechnical earthquake engineering and

while they are used all over the world, in Hungary this research is the first to apply them. It also

has to be noted that most measurement devices use either the Resonant Column or the Torsional

Shear method for testing. In contrast to them, the device used and improved in this study is a

combined one, capable of performing both tests on a single specimen, which makes it rather

unique. The benefit of combined testing is that the soil response can be measured throughout

the whole range of strains from very small strains (10-4 %) up to large strains (10-1 %) on one

single specimen. Analysis of the data and interpretation of results are the most important steps

in the measurement procedure, hence this study focused on both of them. Visual Basic for Ap-

plications in MS Excel has been used to improve measurement result interpretation; the devel-

oped codes are presented in the Annex of the thesis.

Applying a material model to the measured set of data is also an important task and can

be considered as another method of measurement data interpretation. For simple cases, this task

is often done with the trial and error method, while complex models require solving an inverse


5

problem. Some programs used for geotechnical finite element modeling (e.g. Plaxis) provide

built-in tools (Soil Test module) to perform this task for commonly used laboratory tests and

material models, however there is no such tool available for any of the two laboratory measure-

ment methods or any material model which considers the small strain stiffness of soils. There-

fore, in order to obtain material parameters that provide the best fit for a certain model, a pro-

cedure has been developed to solve the inverse problem and provide material parameters by

regression analysis based on measured behavior. For this MS Excel Solver’s Generalized Re-

duced Gradient method was used to minimize the sum of residual squares between values pro-

vided by the model and measured values.

Finally, to preview the solution of the inverse problem, a 3 dimensional finite element

model has been developed and the modeling of several loading stages of the Torsional Shear

Test has been performed using the Ramberg-Osgood material model. Analysis results are com-

pared to measurement results to provide verification of the material model implemented in the

computer program used.

2 Dynamic behavior of soils

A couple of decades ago geotechnical engineers thought that deformation of soils should be

separately assessed for the dynamic and static loading cases. Distinction has been made between

a so-called dynamic and static modulus. It was thought, that the former is only valid for dynamic

calculations and has no relation with the latter, which is used for traditional geotechnical prob-

lems, which consider larger and permanent strains. At that time, soil response for dynamic

loading was thought to be linear elastic.

Based on this approach laboratory observations were used to estimate the ratio of these

moduli for different soil types, see e.g. Figure 2-1. Correlations like this are still popular, since

they seem easy to use; similar graphs can still be found in handbooks, see (DGGT, 2001) or

(Smoltczyk, 2002). However, this method of interpretation of static and dynamic modulus could

lead to serious under- or overestimation of the dynamic modulus; see the comprehensive dis-

cussion by (Wichtmann & Triantafyllidis, 2009).

Later more precise static laboratory measurements revealed that the higher stiffness is not

connected to dynamic loading (i.e. the frequency of loading), but to the achieved strain ampli-

tude. These observations were made first in the 1980s when triaxial laboratory tests were per-


6

formed at Imperial College, London (Burland, 1989), with strains measured locally on the sam-

ple. Earlier the standard testing method suggested to measure strains externally between the end

platens. The concept and basic findings are summarized in Figure 2-2.

Figure 2-1 Ratio of dynamic and static moduli presented by (Alpan, 1970)

Figure 2-2 External and local measurements in a triaxial test: setup (left) and deviatoric

stress-strain curves with zoom into small strains (right) (Puzrin, 2012)

The obtained deviatoric stress-strain curves are close to each other, but when focusing on

the very small strains (up to 0.01-0.1%) the curve of the local strain measurement (dashed) is

highly nonlinear while the curve of the external strain measurement (solid) is almost linear.

Beside this, the initial tangent shear modulus G0 is almost an order of magnitude higher than

the pre-yielding secant modulus, G. They also found that even at small strains, stress-strain

behavior is not entirely reversible; some very small permanent strain was measured (showed by

dashed unloading curve in Figure 2-2). These observations were very important, because in

many practical geotechnical calculations deformations are overestimated. This might be due to

the fact, that in classical geotechnical problems large strains only appear in a relatively small


7

volume of the soil. The strains in the remaining part of the model are very small, but to calculate

displacements, strains are integrated over a relatively large area and the contribution of small

strains can be significant. Therefore, ignoring the stiffer small strain response will lead to over-

estimation of displacements in geotechnical calculations.

As measurement techniques further developed, a transition from the small strain stiffness

or dynamic stiffness and the static stiffness has been observed. Further, more precise laboratory

test results made it clear, that a limit of strains can be found experimentally, above which soils

exhibit damping and non-linear stiffness response, therefore the assumption of linear elasticity

is no longer valid; and that larger strain response is connected through the so-called modulus

degradation or modulus reduction curve, which introduced the concept of strain dependent stiff-

ness.

2.1 Small strain nonlinearity

Recently many studies have shown that the accurate modeling of stiffness degradation with

strain is a key aspect in many dynamic as well as static geotechnical numerical calculations

(e.g. earthquake vibrations, high-speed railway induced vibrations, settlement calculations

around retaining walls or tunnels etc.). It has been recognized, that soils behave linear-elas-

tically only at very low strains (10-6 or 10-4 %) and as the strains (and stresses) increase, the

initially constant shear stiffness or as they often call it in literature, small strain stiffness (Gmax

or G0) decreases gradually, as shown in Figure 2-3, while damping or hysteretic behavior be-

comes more pronounced. Decrease of the stiffness is caused by separation or slippage of inter-

granular contacts as shear strain increases, which means these contacts contribute less and less

to the elastic stiffness of the assembly. However, if cyclic loading is performed, after load re-

versal the reduction is disappears, as previously slipping contacts re-engage in the opposite

direction. Naturally, if loading is continued in this direction, elastic contacts will once again be

lost and stiffness will reduce as before.

Modulus reductions to 5-10% of the original, low-strain value are common. The strain

level, at which this stiffness degradation starts is called threshold level and depends on soil type.

If the strains in the soil increase above this level, non-linear behavior can be observed.

This very small strain level is typical in earthquake induced waves and different kinds of

vibrations. It is well known, that the released energy of the earthquake induces seismic waves


8

that propagate in the rock and soil volume (primary/compression/p-waves and second-

ary/shear/s-waves) and on the ground surface (Rayleigh waves and Love waves), see Figure

2-4.

Figure 2-3 Stiffness degradation of soil with typical strain ranges after

(Atkinson & Sallfors, 1991)

Figure 2-4 Seismic waves (original image by Prof. L. Braile ©)

The most important waves from an engineering point-of-view are shear waves, as they

carry a considerable amount of the energy and they interact with engineering structures in a

Primary, P, Compression, Dilatation Wave Secondary, S, Shear, Distortion Wave

Rayleigh, R, Surface Wave Love, L, Surface Wave


9

very destructive way. These waves tend to refract on layer boundaries and when propagating to

the ground surface, they usually exit stiffer and enter into softer layers; hence they tend to lean

more and more upwards. As a result, when getting in interaction with engineering structures

they are usually already almost vertically propagating, horizontally polarized shear waves;

which induce horizontal shear strains in the soil, as shown in Figure 2-5. This is one of the main

reasons that the description of small strain stiffness is approached usually by introducing shear

modulus and shear strain level.

Figure 2-5 Propagation of shear waves in layered medium and induced deformation

A detailed discussion about the small strain stiffness of soils is presented in the Rankine

lecture of (Clayton, 2011) where first the historical development of elastic theory and the con-

stitutive frameworks for its use in soil modeling is elaborated. Afterwards it is shown, how the

comprehensive theoretical description of anisotropic elasticity can be simplified for practical

calculations regarding soils. It should be emphasized, that beside anisotropy, several other fac-

tors have an effect on both the small strain stiffness and the stiffness at larger strains.

The most important factors that affect small strain stiffness are:

void ratio

grain properties (grain size and shape)

effective overburden stress

stress history

rate of loading

t

g

vertically propagating, horizontally

polarized shear waves

shear strain

shear stress

vs=200 m/s

vs=400 m/s

vs=700 m/s

vs=1000 m/s

vs=1500 m/s

vs=2000 m/s

vs=2500 m/s


10

structure and fabric of soil

discontinuities

Factors that control the degradation of stiffness at larger strains are:

strain level

loading path (change in effective stress)

destructuring

change in rate of loading

Some of these factors can be assessed by detailed laboratory investigations, however in

many practical cases there is no room for such elaborate testing, hence simplifications are made

i.e. some factors are neglected. As (Clayton, 2011) discussed it, anisotropy may have a signifi-

cant effect in overconsolidated soils. Rate effects are now considered relatively unimportant at

very small strain levels. At higher strain levels, however three main effects connected to strain

rates can be observed. The first one is that the elastic range, where small strain stiffness can be

observed increases with strain rate. Because of this tests with different strain rate (e.g. resonant

column and cyclic triaxial tests) tend to show different stiffness degradation at small strains.

Second, shear stiffness depends on the strain rate the most at intermediate strains, i.e. between

10-2 % and 10-1 % strain. Last, tests with higher strain rates (e.g. resonant column) are likely to

provide a lower limit for the degradation curve when compared to other test methods with lower

strain rates. The effects of structure and fabric are more difficult to describe, for soils exhibiting

these features, detailed field investigations can be suggested. The remaining factors can be in-

vestigated in laboratory in detail and several approaches have already been presented for spe-

cific soils in literature for handling them, see (Kramer, 1995), (Benz, 2006) and (Clayton, 2011).

Naturally, the three-phase model for soils, where the soil mass is considered to consist of

solid particles, groundwater and air in the voids, can be applied within this modeling task too.

A distinction should be made between the ‘undrained’, ‘short term’ or ‘end of construction’

case and the ‘drained’, ‘long term’ or ‘total stress’ case. It should be noted that the shear stiff-

ness remains the same in the undrained and drained cases, since shearing usually involves

change in shape without change in volume and the contribution of water to shear stiffness is

negligible at low rates of strain. However, other elastic parameters (Young’s modulus, Pois-

son’s ratio, bulk modulus) depend significantly on whether the effective or total stress approach

is used for their determination.


11

2.1.1 Modeling small strain nonlinearity

Several analytical functions have been considered for curve-fitting the deviatoric stress-strain

behavior of soils at small strains in literature. The two most often mentioned are the hyperbolic

and the Ramberg-Osgood function; see (Puzrin, 2012). Because nonlinearity is most dominant

in the small strain range the normalization of stress-strain curves is often used in these models

(Puzrin, 2012). After reaching a limiting deviatoric stress qL the stress strain curve can be ap-

proximated well enough with by a straight line with the inclination 3GL, which is then followed

up to the yield stress qy. GL is the tangent shear modulus corresponding to qL. Then normaliza-

tion of the curve by qL is done and an analytical function is used to describe the behavior up to

qL, this is shown in Figure 2-6.

Figure 2-6 Deviatoric stress-strain curve: true (left) and normalized (right) (Puzrin, 2012)

For the normalization, the following notation is used:

y = q

qL , x = εs

εsr , εsr =

qL3G0

, xL = εsL

εsr Equation 2-1

where y and x are the normalized deviator stress and strain, respectively; G0 is the initial tangent

shear modulus; xL is the normalized limiting shear strain. A similar normalization is used with

the hyperbolic function, which is implemented into the Hardening Soil model where qL is taken

as 90% of qy (Benz, 2006). The way this normalization is used will be shown more in detail for

the Ramberg-Osgood model in Chapter 2.1.4.

As mentioned before in most nonlinear cyclic stress-strain models the relationship be-

tween the shear stress and shear strain is defined; and this is usually done by setting up two

basic functions: an initial loading curve and a hysteresis loop. The initial nonlinear loading

curve can be expressed in the form:


12

τ = f(γ) Equation 2-2

where t is shear stress and gis shear strain.

This relationship is usually obtained from some kind of monotonic loading test. If the

load at a given (tp, gp) turnaround point is then reversed, the formulation of the hysteresis loop

is used, which is derived from the original function as follows:

τ – τp

2 = f (

γ – γp

2) Equation 2-3

This implies that the curve of the unloading and opposite direction loading is the same

shape as the initial loading curve, but it is stretched by a factor of 2, see Figure 2-7. This is

because previously slipping contacts must first recoil elastically until they are unloaded. Then,

if the loading in the opposite direction is continued, they will distort backward by the same

amount. After a second load reversal at this second turnaround point (–tp, –gp), the hysteresis

curve is formed. This third part of the curve can be calculated by:

τ + τp

2= f (

γ – γp

2) Equation 2-4

Figure 2-7 Nonlinear stress-strain curve for initial loading and hysteresis curve for cyclic

loading

The hysteresis loop is formed with always keeping track of the last turnaround point

which will be the origin for the next half loop. This means during cyclic harmonic loading the


13

origin of loading will not be reached any more. Each point of the initial loading curve (shown

dashed in Figure 2-7) can be considered to be a turnaround point of a smaller hysteresis loop

and it is also called backbone curve or skeleton curve (Ishihara, 1996). The rule of formulating

the hysteresis curve as described above is called the Masing rule. A more detailed elaboration

about this is given in Chapter 2.1.3.

Hysteretic behavior is caused by energy dissipation i.e. damping. To describe damping it

is usual to assess the energy that is lost during one load cycle. This is equal to the area enclosed

by the hysteresis loop. Energy loss is then usually normalized by the elastic stored energy, for

a detailed description see (Ishihara, 1996). Similarly to stiffness, damping is also depending on

strain level, but an opposite tendency can be observed, namely damping increases with strain

amplitude. This tendency is usually captured by the damping curve.

Several complex soil models have been developed to capture the nonlinear cyclic behav-

ior of soils. In general, these models define a stiffness degradation curve and a damping curve,

and they can only produce accurate results, if sophisticated laboratory and field test are used to

determine their key model parameters. (Kramer, 1995) categorized the models into three clas-

ses:

equivalent linear model

cyclic nonlinear models

advanced constitutive models

Of these the first is the simplest but most frequently used in geotechnical earthquake en-

gineering for ground response analysis, although it is not capable to model the actual path of

the hysteresis loop. The last class contains complex material models which are used in finite

element software and can describe several important aspects of soil behavior (e.g. initial stress

conditions, different stress paths, rotating principal stresses, cyclic loading, different strain

rates, and undrained condition). Due to this they usually require many model parameters and

their careful calibration. Models in the middle class focus on the description of hysteretic be-

havior and strain dependent modulus and damping. In the following the equivalent linear and a

cyclic nonlinear model, the Ramberg-Osgood model are presented. The practical formulation

of the Ramberg-Osgood model is shown detailed, because it will be used later in this study.

2.1.2 Equivalent linear model

This model uses a linear calculation approach and iteration to account for the strain dependent

stiffness. For very small shear strains soils usually respond according to linear elastic behavior.


14

As discussed in (Kramer, 1995) based on Hooke’s law, it is possible to relate the elastic prop-

erties of soils and the wave propagation velocities:

vp= √G (2 – 2ν)

ρ (1 – 2ν) Equation 2-5

where vp is p-wave velocity, G is shear modulus, is Poisson ratio, is density.

vs=√G

ρ Equation 2-6

where vs is s-wave velocity.

In these equations from a practical point of view, since induced strains are usually very

small, G can be considered to be Gmax. However, at larger strains, elasticity is no longer valid.

To distinguish the strain range of these two behaviors the concept of linear cyclic threshold

strain was introduced in (Vucetic, 1994). If the induced shear strains are larger than the thresh-

old strain, a cyclic loading pattern will result in a hysteretic stress-strain response, as shown in

Figure 2-7. The description of the behavior is detailed more in Figure 2-8, which introduces

two shear moduli.

Figure 2-8 Secant shear modulus and tangent shear modulus

In each point of the loop a tangent can be obtained, the slope of which defines the actual

shear stiffness, usually called tangent shear stiffness Gtan. For describing one full cycle the stiff-

ness can be defined as the secant shear stiffness Gsec which can be considered the average stiff-

ness during the whole cycle:

(tp, gp)

t

g

Gtan

(-tp, -gp)

Gsec


15

Gsec=τp

γp

Equation 2-7

where tp and gp are shear stress and shear strain at the peak respectively. This model only uses

the Gsec modulus to calculate the total shear strain induced by a given shear stress and therefore

it is not capable of modeling each point of the cycle. Instead of modeling the whole nonlinear

cycle, the “average” of the response is obtained by a linear calculation, hence the name equiv-

alent linear model. While this is an approximation, this model can still produce acceptable re-

sults e.g. in ground response analysis with considerably faster calculation time.

The breadth of the hysteresis loop or more precisely the area within the loop is connected

to the energy dissipation during that loop and it can be quantified by the damping ratio D:

D = WD

4π WS =

1

4π

Aloopτp γp

2

= 1

2π

Aloop

Gsec γp 2 Equation 2-8

where WD is the dissipated energy, WS is the maximum strain energy of the equivalent elastic

material, and Aloop is the area of the hysteresis loop.

Figure 2-9 Hysteresis loop of one cycle showing dissipated energy WD, and equivalent elastic

strain energy WS

Energy dissipation occurs mainly due to friction between particles and as a consequence

due to heat emission. If the loading is repeated with different amplitudes, different hysteresis

loops can be obtained, see Figure 2-10. Two observations can be made. The larger the induced

strain, the more inclined to loop gets; hence the secant modulus gets smaller with increasing


16

strain. This is how the stiffness degradation shown in Figure 2-3 can be observed in a test.

Second, the larger the strain, the wider the loop gets; therefore, damping gets larger with in-

creasing strain. The stiffness degradation can be characterized by the degradation curve or by

the backbone curve, which can be obtained by connecting the peaks of hysteresis loops at in-

creasing strain.

Figure 2-10 Hysteresis loops at different strain levels due to harmonic cyclic loading

When trying to describe the nonlinear cyclic behavior of soil with a material model, it is

clear that a linear model is not suitable for precise calculations. However, if a stiffness degra-

dation curve is defined in the model, an iterative approximate solution can be developed which

produces acceptable results. Initially the soils are assigned a stiffness of Gmax. Following each

round of computations, shear strain histories are tabulated for every element. Based on those

histories, new reduced G values are assigned to each element. The response is recalculated and

the new strain histories are compared to the previous ones. If the difference between them is

τ

t

τ

t

τ

t

τ

t

0

5

10

15

20

25

30

35

0.0001 0.001 0.01 0.1 1 g[%]

Gse

c[M

Pa]

Gmax

τ

g

Gsec

τ

g

τ

g

τ

g


17

less than a prescribed error value, the analysis is finished. If not, a new stiffness is determined

and response is calculated again. This method usually converges after 4 or 5 iterations.

Since linearization is just an approximation of real soil behavior, there are drawbacks.

The reduced modulus is applied for the entire duration of shaking, giving a very poor simulation

of high frequency response and allowing soft layers to absorb too much energy. No permanent

deformations can be calculated with this model, after unloading the strain will always return to

zero. It is also important to note, that failure cannot be modeled with this approach either, be-

cause it has no limiting strength.

2.1.3 Modeling irreversible small strain behavior

A simplified approach for considering irreversible strains which are often used for extending

the nonlinear small strain models is the deformational plasticity (Puzrin, 2012). For this the so

called Masing rules have to be introduced which were first published in (Masing, 1926) to de-

scribe the behavior of brass under cyclic loading and have been since adopted to soils and ex-

tended as well (Pyke, 1979). In (Benz, 2006) the following comprehensive summary is given

about them:

1. The shear modulus in unloading is equal to the initial tangent modulus for the initial load-

ing curve.

2. The shape of the unloading and reloading curve is equal to the initial loading curve, except

that its scale is enlarged by a factor of two in both directions.

The first two rules are sufficient to describe regular, symmetric cyclic loading. If the

loading is more general or irregular, meaning not symmetrical or periodic, the following two

additional rules are commonly added to the two original Masing rules:

3. Unloading and reloading curves should follow the initial curve in case the previous maxi-

mum shear strain is exceeded.

4. If the current loading or unloading curve intersects a previous one, it should follow the pre-

vious curve.

The four rules together are known in literature as the extended Masing rules. Figure 2-11

shows hysteresis loops in symmetric and irregular loading that comply with these four rules. It

should be noted that Rule 4 has been a topic of discussion in literature, see (Benz, 2006). Figure

2-11 (b) shows three possible paths (A, B and C) the curve could follow after unloading and

reloading within a previous larger loop.


18

Figure 2-11 Hysteresis loops in symmetric (a) and irregular loading (b) according to the ex-

tended Masing rules (redrawn from (Benz, 2006), after (Pyke, 1979))

2.1.4 Modified Ramberg-Osgood model

The Ramberg-Osgood model originally was introduced in (Ramberg & Osgood, 1943). The

proposed model used three parameters for describing stress-strain curves of aluminum-alloy

and steel sheets. Later in (Idriss, Dobry, & Singh, 1978) an adaptation of the model was pro-

posed in order to obtain the shear modulus reduction. Since then it was used in several studies

to model small strain soil behavior, e.g. (Ray & Woods, 1988).

In general form, the proposed function for the normalized stress-strain relationship intro-

duced in Figure 2-6 and its first derivative are:

x = y + α y R and dy

dx =

1

1+α R y R–1 Equation 2-9

where parameters

α = xL – 1 and R = 1–b

(xL –1) b Equation 2-10

are chosen to fit curves obtained by laboratory measurements.

The derived formulation to be used in practical calculations is detailed in the following.

Shear stress-strain relation is described by a hyperbola:


19

γ = τ

Gmax(1+α |

τC τmax

|R–1

) Equation 2-11

where g and t are shear strain and stress, Gmax is the small strain shear modulus, tmax is maximum

shear stress, C and R are model constants. Note that in this formulation shear stress is nor-

malized by the maximum shear stress which is usually obtained from triaxial test results. The

stress-strain formulation is straightforward, although it is very difficult to invert. Figure 2-12

shows typical stress strain curves obtained with the model.

Figure 2-12 Stress-strain curves obtained with the Ramberg-Osgood model

Definitions for secant and tangent modulus follow directly from their definitions such

that:

Gsec= τγ

= G

(1+α|τ

C τmax|R–1

)

Equation 2-12

and for tangent modulus:

Gtan= ∂τ∂γ

= G

(1+αR|τ

C τmax|R–1

)

Equation 2-13


20

Secant modulus is often used for equivalent linear ground response analysis calculations,

while tangent modulus is used in calculations which use time-stepping such as the method of

characteristics.

The degradation curve for secant modulus is given by:

Gsec

Gmax =

1

1+α|τ

C τmax|R–1 Equation 2-14

Plots of normalized secant and tangent modulus vs. strain are given in Figure 2-13. Note

that the computation of moduli vs. strain is not a straightforward process, since strain does not

appear in the formulation of the degradation curve. However, one may choose a value of t, and

then compute both moduli with Equation 2-12 and Equation 2-13 then compute strain via Equa-

tion 2-11. This can easily be done with spreadsheets.

Figure 2-13 Normalized secant and tangent modulus vs. strain obtained with the Ramberg-

Osgood model

Some obvious behavior from Figure 2-13 is worth noting. First; strain is plotted on the

horizontal axis using a logarithmic scale. This shows typical soil behavior more clearly than a

linear scale would and so it is often done in literature. One has to keep in mind that strain is

often expressed in percent (%) instead of (mm/mm) and is generally single amplitude, peak

value. Some laboratory testing will measure RMS value (0.707x single amplitude) or peak-to-

peak (2x single amplitude). Another treatment of strain occurs when equivalent linear models

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.0001 0.001 0.01 0.10

G/G

max

Shear strain g [mm/mm]

Gsec/Gmax

a=1.0,C=0.8,R=2.5

Gtan/Gmax

a=1.0,C=0.8,R=2.5


21

(e.g. SHAKE, Strata, FLUSH) are used and an effective strain (γeff = 0.65x single amplitude)

may be used. These are very common blunders in transposing various authors’ data from one

graph to another. Second; tangent modulus will always be less than secant modulus at any given

strain. Third; modulus ratios become significantly less than 1.0 at fairly low strain levels (g =

0.001). Finally; soils can behave significantly different from what is shown here. These curves

can slide horizontally by an order of magnitude in either direction or change shape.

The dimensionless form allows for comparing test data for the same soil at different con-

fining stresses. These data then can be applied to the entire soil layer while considering the

effects of confining stress. The dimensionless form of the formulation follows as:

γ

γr =

τ

τmax(1+α |

τ

C τmax|R–1

) Equation 2-15

where gr is reference shear strain at tmax shear stress:

γr =

τmax

G Equation 2-16

Using dimensionless values is not difficult; however, some agreement must be made

about the evaluation of tmax for each specific soil layer. A typical assumption is Mohr-Coulomb

failure envelope strength with effective stress properties. Such an envelope is shown in Figure

2-14. Based on the diagram the maximum shear strength is:

τmax= (c'

tanφ' +σ0' ) sin φ' cos φ' Equation 2-17

τmax=√((1+K0

2) σv

' sin φ'+c' cos φ')

2

– ((1–K0

2) σv

' )

2

Equation 2-18

Figure 2-14 Mohr diagram for maximum shear stress


22

The value of maximum strength will vary with average confining stress ’0. Dimension-

less stress-strain curves can be generated in the same manner as the curves in Figure 2-12 by

simply substituting values into Equation 2-15. The dimensionless strain values are different

than “regular” strain values shown in Figure 2-12 by a factor of about 1000. While this may

first seem to be a waste of effort, when viewed from the perspective of laboratory data, it is

very valuable since specimens under different confining stress will exhibit different reduction

curves. The process of normalizing collapses the curves onto one common curve if they are

similar soils. The curves can then be reconstructed for any confining stress or depth of interest,

by reversing the process.

The formulation by Midas (MIDAS Information Technology Co., 2014) is slightly dif-

ferent than the one presented above, and it is shown here because this notation is used in Chapter

3.5. The main equation for initial loading is given as:

G0 γ = τ + α |τ|β τ Equation 2-19

where G0 is the small strain stiffness (shear modulus), g and t are shear strain and shear stress

respectively, and are model parameters given by:

α = (2

γr G0

)β

and β =2π hmax

2 – π hmax Equation 2-20

where gr is the reference shear strain and hmax is the maximum damping constant.

For unloading and reloading the hysteresis curve is as follows:

G0 (γ ± γ1

2) = (

τ ± τ12

) (1+α (ττ ± τ1

2)

β

) Equation 2-21

where g1 and t1 are the shear strain and stress values at the turnaround point.

2.2 Measurement of dynamic soil properties

The assessment of dynamic soil behavior presents quite a big challenge for geotechnical engi-

neers due to numerous reasons. First of all, the loads to be considered are quite different than

the usual loads civil engineers need to consider. To be precise the loading itself is dynamic (or

cyclic) and not the soil behavior or soil properties, however since these expressions are widely

used in literature, they are used in this study also. Most dynamic problems are connected to

some kind of wave propagation (e.g. vibrations of machine foundation, cyclic loading of waves

in water or earthquake loading) and it is already difficult to describe and quantify the loading.

Secondly, these loads can induce strains ranging from a very low level (~10-4 %) to a high level


23

(~1 %). As shown in Chapter 2.1 the governing soil properties for very low strain problems are

stiffness and damping. In some cases of large strains, the failure of soil masses has to be as-

sessed as well; hence the strength has to be determined too.

The behavior of soils subjected to dynamic loads is governed by dynamic soil properties.

There are many tests used for determining parameters, which are describing soil behavior.

These can be organized into two main types, field tests and laboratory tests. There are ad-

vantages and disadvantages to both groups. Field tests are particularly useful, because they are

commonly used for examining a bigger section of the subsoil in contrary with laboratory tests,

where only a number of smaller samples are studied and their properties are then carefully used

to describe a larger part of the subsoil. There are of course exceptions, e.g. pressuremeter test-

ing, where installation and bedding effects can still be significant, because here a relatively

small soil surrounding is tested close to the pressuremeter. The main disadvantage of field tests

is that can only be used to monitor the soil behavior in the in-situ state, no changes of state, e.g.

change of water content, void ratio can be studied. Laboratory tests can naturally be used for

such assessments; although for those the effects of sampling and disturbance during sample

preparation make the examinations difficult. The disturbance usually appears through the

change of the in-situ stress state (air entry, swelling, destructuring). The selection of testing

method should be based on considerations of initial stress state, stress history and the antici-

pated dynamic loading conditions of the specific problem as well as the variability of the

ground. If the variability of the ground is much higher in the vertical direction than the hori-

zontal, as in most cases, basic soil testing can be used to identify main layers (profiling) and the

determination of stiffness and stiffness degradation can be focused on samples taken from these

layers. However, if lateral continuity cannot be established, the priority is to perform in situ

tests, which describe the behavior of the larger area (e.g. MASW profiling) or use more regular

in situ tests (e.g. CPT) and approximate correlations based on them.

2.2.1 In-situ tests

Field tests can be categorized into groups based on different aspects; whether they are per-

formed on the surface or not, whether they induce low strains or larger strains etc. Seismic

geophysical tests are an important group of field tests. They are usually performed by inducing

transient or steady state stress waves and the propagation of them are then observed at different

locations. These tests are particularly advantageous because they induce similar soil defor-

mations that are induced by the considered dynamic load e.g. earthquake loads or waves gen-

erated by machine foundations. In a noisy environment, the induced small deformations can be


24

difficult to distinguish from other deformations originating from traffic or other man-made vi-

brations. In such cases a number of measurements are ‘stacked’ i.e. added and then random

noise portions of the obtained signal tend to cancel each other while the actual induced waves

are magnified. Many traditionally used in-situ soil investigation methods induce large strains

in the subsoil (e.g. CPT, SPT, DP, pressuremeter etc.). Although these tests cannot be used

directly for determining dynamic soil properties, their results have been correlated to low-strain

soil properties by numerous authors, e.g. (Wolf & Ray, 2017), so they can be used for a first

estimate. The most commonly used field tests are discussed shortly in the following. A common

feature is, that they all are based on the measurement of the shear wave velocity and they can

only be used for determining the small strain stiffness Gmax, so no damping or modulus reduc-

tion properties can be obtained with them. The choice of method depends on the geologic pro-

file, the maximum profiling depth, the size of the investigated site and the importance of the

structure or facility. In many cases the field investigation is limited to the top 30 m of the soil

layers in order to assess the site based on code regulations (e.g. soil classes in Eurocode 8).

Cross hole seismic test

Cross hole testing is performed within two or more PVC-cased vertical boreholes. The contact

between casing and soil can significantly affect the measurement; therefore, usually grouting is

used to reduce voids outside the casing. The quality of the contact can be checked with sonic

logging methods prior to the seismic test. During the test itself the velocity of two types of

horizontally travelling seismic waves can be measured; primary compression (P-wave) and sec-

ondary shear (S-wave) waves. In one borehole, a wave generator is used as a source at the test

depth and in another borehole (or more boreholes) velocity transducers are used as receivers in

the same depth. Both source and receiver are usually fixed in position by pneumatic packers

pushing them to the casing. Before testing the verticality of the casing is established with incli-

nometers to obtain the exact distance between source and receiver. Wave propagation velocity

is obtained by measuring the travel time and using the travel distance. Data acquisition is usu-

ally done by digital systems, evaluation is based on first arrival times or interval arrival times

over the measured distance. From the measured velocities in situ maximum shear modulus can

be calculated. Soil density is often estimated only from other investigations. Cross hole tomog-

raphy can also be used instead of regular cross hole testing. In this case, arrays of receivers are

used and therefore multiple paths can be recorded for each signal. Then tomographic processing

consisting of inversion is performed (McMechan, 1983) and more precise stiffness profiles can

be obtained.


25

Down hole seismic test

In the seismic down hole test a surface source is used to induce shear waves, usually a sledge-

hammer striking a metal object pushed into the soil and weighted. A receiver is located in a

cased borehole that can be moved to different depth to obtain a shear wave velocity profile. The

source is usually offset from the top of the borehole in order to avoid vibration energy travelling

down the casing; the travel distance has to be determined accordingly. It is beneficial if the

receiver consists of two sets of three orthogonally oriented geophones separated by a known

distance. This way the travel time can be measured between them and a more precise measure-

ment can be made which corresponds to the actual depth and not to all layers above. For as-

sessing the travel time, either P or S waves can be used. For P waves, usually the first arrival is

assessed, while for S waves the peak to peak method is used. Material and radiation damping

reduce wave amplitudes and make measurements below 30-60 m unfeasible.

Suspension PS logging

This method is used in an uncased borehole with use of drilling fluid and its main advantage is

that it can be used to obtain both P and S wave velocity data down to great depths (several

hundred meters). A 6-7 m long probe containing a source and two receivers is lowered down

the borehole suspended by a cable. At the specific test depth, the source generates pressure

waves in the drilling fluid which are converted to P and S waves at the borehole wall and travel

in the surrounding soil towards the receivers where they convert back to pressure waves coming

back to the fluid and finally they reach the receivers. To enhance identification of arrival of P

and S waves the test is repeated with an impulse of opposite polarity. Disadvantage of the

method is, that the induced S waves have a much higher frequency (0,5-2,0 kHz) than those of

interest in geotechnical earthquake engineering.

Seismic cone penetration test (SCPT) and seismic flat dilatometer (SDMT)

These tests are very similar to the down hole test except that there is no need for a borehole

which makes them faster and cheaper. The SCPT is performed in the same manner as the regular

CPT with the addition of usually two geophones or accelerometers located in the CPT tip. The

S and P wave velocity is measured at selected intervals (typically 1 to 2 m) by striking a steel

or wood beam pressed firmly against the ground and calculated based on the difference in travel

time between the consecutive geophones at a given depth. The benefit of the SCPT is that reg-

ular CPT data can be used for soil classification, typically based on interpreted Soil Behavior

Type, and determining other related parameters such as ground water table, density etc.

(Robertson & Cabal, 2014). The SDMT test is also similar to the regular flat dilatometer test


26

with a down hole like wave velocity measurement extension. Advantage of the SDMT test over

the SCPT is, that the penetration of the dilatometer induces somewhat smaller disturbance

(compaction) of the soil while the CPT tends to induce large strains while penetration which

changes the state of the soil prior to the low strain seismic testing.

Spectral analysis of surface waves (SASW) and multichannel analysis of surface waves

(MASW)

Both methods can be performed on the ground surface without boreholes and are based on the

ideas behind the seismic refraction test. They both use the dispersive characteristics of surface

waves to determine the variation of shear wave velocity with depth. The earlier method, the

SASW uses a source (hammer weight or explosive) and two receivers, later researchers devel-

oped the method into the MASW which uses a set of receivers (24 or more) deployed along a

line at regular intervals. The maximum depth of investigation with the SASW can even be sev-

eral km, while the MASW is used for more detailed investigation of the top 30-50 m of soils.

The maximum reachable depth is depending on site and source conditions and is dictated by

the longest wavelength made by the impact source. Greater impact power translates to longer

wavelengths and deeper sampling depths. Vertical low-frequency geophones (<4.5Hz) are rec-

ommended as receivers. The length of the receiver spread usually limited to 50-100 m and it is

directly related to the longest wavelength detected while receiver spacing (distance between

receivers) relates to the shortest wavelength detected. The source and receiver spread distance

is one of the variables that affect the horizontal resolution of the dispersion curve. For MASW

different types of waves are recorded through the multichannel array. The dispersive nature of

different types of waves is imaged through wave-field transformation of seismic record by fre-

quency wavenumber or slowness-frequency transform. From the dispersion image, a dispersion

curve of the fundamental mode of Rayleigh waves is selected, which is then inverted for a vs

profile.

2.2.2 Laboratory tests

The following laboratory tests can be performed on soil specimens which are obtained from

either undisturbed samples or from reconstructed disturbed samples. Undisturbed samples can

be produced by borings and sampling devices of high standard. In many cases their use is un-

feasible and reconstituted samples have to be used. Naturally, there is a difference in soil fabric,

structure, anisotropy etc. between natural and reconstituted specimens; hence there is always

an inherent uncertainty in these tests. However, if a careful and thorough testing program is


27

performed the effect of the differences can be assessed, and test results on reconstituted samples

can be used for a wide range of applications.

Resonant column test

The resonant column test is the most commonly used laboratory test for determining the small

strain stiffness and damping properties of soils. After a cylindrical soil sample is prepared and

consolidated, a cyclic torsional load with very small strain amplitude is applied to the sample

by an electromagnetic loading system. The response of the sample is measured by an accel-

erometer. The frequency of the harmonic load is changed until the maximum strain amplitude

response is reached, which corresponds to the fundamental or resonance frequency. The shear

modulus can be obtained by assessing the geometry of the specimen and certain characteristics

of the device. Some devices use a hollow cylinder, which is beneficial because in such a sample

the variation of shear strain and stress in the sample induced by the torsional load is smaller

than in a full cylinder. Advantage of the measurement is that the strain amplitude examined

with it is in the small strain range (10-4 %) and effect of confining pressure, void ratio and time

can be investigated. However usually only isotropic stress states can be investigated and pore

water pressure is difficult to measure. Since this thesis focuses on resonant column and torsional

shear testing the detailed description of the test is given in Chapter 3.

Bender element

Bender elements can be used for determining shear wave velocity of soil samples. This method

is often used in combination with a more sophisticated laboratory test, e.g. cyclic or regular

triaxial testing, resonant column or oedometer test. The bender element itself consists of small

piezoelectric plates (in the cm range), which are attached to the base and top plates used in the

test, protruding into both ends of the cylindrical sample. They are capable of inducing a con-

trolled movement when an electrical signal is imposed on them. Usually two plates are used so

that P and S waves can also be generated by their coupled movement. Some piezoelectric crys-

tals can also be used as receivers. This means they induce an electrical signal if they undergo a

certain deformation. Based on sample size and recorded travel time of the induced seismic

waves, shear wave velocity can be obtained. Several wave types are used (single sine, sine

sweep etc.) for the test. Data analysis and interpretation can be quite difficult and is still the

topic of research, see e.g. (Szilvágyi, Panuska, & Ray, 2018).

Cyclic triaxial test

The regular triaxial device can be supplemented with a loading piston capable of cyclic loading

and be used for obtaining dynamic soil properties at high strain levels. In this test, a cylindrical


28

specimen is tested in a triaxial cell. The specimen is surrounded with a rubber membrane and a

top cap with a porous stone is attached to the top of it. The cell is either filled with water or just

with air and the pressure in the cell can be regulated while a vertical loading piston is connected

to the top cap. This setup allows for simulating axisymmetric stress states and complex stress

paths. Drained and undrained tests can also be run. For the cyclic test, usually the deviator stress

is changed harmonically with time and pore pressure increase due to the cyclic load is carefully

logged. This type of testing is widely used for assessing liquefaction potential. The biggest

advantage is the wide range of anisotropic stress states that can be simulated; however, shear

stress conditions during seismic wave propagation cannot be modeled.

Cyclic direct simple shear test

This test is capable of simulating shear stresses in a soil sample which are acting during earth-

quakes much better than the cyclic triaxial test. The most important waves of an earthquake

from a geotechnical engineering point of view are vertically propagating, horizontally polarized

shear waves (Figure 2-5) which can be simulated in this test. A short cylindrical sample is

loaded vertically by a constant normal pressure while a horizontal cyclic loading is also applied.

The sample is restrained against lateral expansion by a series of rings, a wired membrane or

rigid boundary platens. Since the shear stress is applied at the top and bottom of the specimen

this load will induce non-uniformly distributed shear and normal stresses in the sample. Sample

height enlarges this effect, therefore short samples are recommended (diameter/height ratio

greater than 8:1).

Cyclic torsional shear test

This test also uses cylindrical soil samples enclosed in a rubber membrane. The horizontal cy-

clic load is applied on top of the specimen in a torsional manner, similarly to the resonant col-

umn test. Some devices are capable of imposing a vertical load on the top of the sample also

and this allows for testing under anisotropic stress conditions. The test can be used to obtain

shear stiffness and damping over a wide range of strain levels. Some devices use hollow cylin-

drical samples which allow for an almost uniform distribution of shear stress and strain in the

sample. This has an advantageous effect on precision but at the same time it makes sample

preparation quite difficult.

2.3 Determination of dynamic soil parameters based on correlations

In the following, a literature review is presented for obtaining dynamic soil properties of sands

with correlations. These correlations can be used for initial estimates in case there are no field


29

or laboratory measurement data available or to check the order-of-magnitude of experimental

measurements. They were obtained by fitting equations to laboratory test results. They are also

often used for normalization of laboratory measurement results, to be able to compare them to

other researcher’s findings without the need to perform the tests under the exact same condi-

tions.

2.3.1 Small strain stiffness

The most common correlations for Gmax include the use of a void ratio function and a confine-

ment stress component. A well-known general equation was introduced by (Hardin & Richart,

1963), (Hardin & Black, 1966) which correlates shear modulus in [MPa] to void ratio and con-

finement pressure as:

Gmax=A F(e)p' n

Equation 2-22

where A and n are experimentally found coefficients, p’ is the mean effective stress in [kPa]

and F(e) is the function of void ratio usually taken as:

F(e) = (2.17 – e)

2

1+e Equation 2-23

It should be noted that form of F(e) varies from researcher to researcher and the constant

2.17 can be treated as an additional coefficient a so that:

F(e) = (a – e)

2

1+e Equation 2-24

The dimensionless form of Equation 2-22 is also frequently used with normalized pres-

sure:

Gmax=A F(e) (p'

patm)

n Equation 2-25

where the use of atmospheric pressure patm allows for any convenient units of measure. Equation

2-25 is the most often used correlation for estimating Gmax and the coefficients A, a and n are

also called intrinsic (state-independent) parameters associated with small-strain stiffness

(Carraro, Prezzi, & Salgado, 2009).

A summarizing table about many author’s correlations for Gmax values (in MPa) of cleans

sands using Equation 2-25 (with p’ in [kPa]) was presented in (Benz, 2006) and is shown in

Table 2-1.


30

From Table 2-1 it is clear that many tests have been made on sands, although most of

them are poorly graded clean sands; which means sands without fines and a maximum uni-

formity coefficient CU of 2. Researchers early on observed, that the gradation curve of sands

have a significant effect on Gmax. (Iwasaki & Tatsuoka, 1977) stated that shear modulus de-

creases with increase of uniformity coefficient and also decreases with the increase in the con-

tent of fine particles FC; therefore, the correlations obtained for clean sands usually overesti-

mate Gmax of sands with a higher CU and sands with fines content. To account for the effect of

fines and higher CU (Iwasaki & Tatsuoka, 1977) introduced an additional variable B to Equation

2-22 as a multiplier smaller than 1.0, although they did not specify a way to determine its value

based on a given CU and FC. Figure 2-15 shows obtained values for B.

It should be noted that FC in this paper meant mass % of particles smaller than 0.074 mm

and not the usual 0.063 mm. Furthermore, the use of the reduction factor B is difficult, because

it is not specified, how to obtain B if both CU and FC has to be accounted for. Reduction values

of 60-90% were measured for CU values of 2-5. A FC of 5% means a reduction of Gmax to 70%

which is significant.

Figure 2-15 Reduction factor B to account for higher CU and FC obtained

by (Iwasaki & Tatsuoka, 1977)


31

Table 2-1 Correlations for shear modulus Gmax of clean sands based on (Benz, 2006)(using

Equation 2-22 for Gmax in [MPa] with p’ in [kPa])

Soil tested d50

[mm]

CU

[-]

A

[-]

F(e)

[-]

n

[-]

Reference

Kenya car-

bonate sand 0.13 1.86 101-129 e-0.8 0.45-0.52 (Fioravante, 2000)

Toyoura sand

(subangular) 0.16 1.46 71-87

(2.17-e)2

1+e 0.41-0.51

(Hoque & Tatsuoka,

2004)

Toyoura sand

(subangular) 0.19 1.56 84-104

(2.17-e)2

1+e 0.50-0.57

(Chaudhary, Kuwano, &

Hayano, 2004)

Silica sand

(subangular) 0.20 1.10 80

(2.17-e)2

1+e 0.5

(Kallioglou,

Papadopoulou, &

Pitilakis, 2003)

Silica sand


(2.17-e)2

1+e 0.5

(Kallioglou,

Papadopoulou, &

Pitilakis, 2003)

Silica sand


(2.17-e)2

1+e 0.5

(Kallioglou,

Papadopoulou, &

Pitilakis, 2003)

Toyoura sand

(subangular) 0.22 1.35 72 e-1.3 0.45

(Lo Presti, Pallara,

Lancellotta, Armandi, &

Maniscalco, 1993)

H.River sand

(subangular) 0.27 1.67 72-81

(2.17-e)2

1+e 0.50-0.52

(Kuwano & Jardine,

2002)

Glass ballotini

(spheres) 0.27 1.28 64-69

(2.17-e)2

1+e 0.55-0.56

(Kuwano & Jardine,

2002)

Hostun sand

(angular) 0.31 1.94 80

(2.17-e)2

1+e 0.47

(Hoque & Tatsuoka,

2000)

Silica sand

(angular) 0.32 2.80 48

(2.17-e)2

1+e 0.5

(Kallioglou,

Papadopoulou, &

Pitilakis, 2003)

Ticino sand

(subangular) 0.50 1.33 61-64

(2.17-e)2

1+e 0.44-0.53

(Hoque & Tatsuoka,

2004)

Ticino sand


(2.27-e)2

1+e 0.43



Maniscalco, 1993)

Silica sand 0.55 1.80 275 (2.17-e)

2

1+e 0.42

(Wichtmann &

Triantafyllidis, 2004)

Ticino sand

(subangular) 0.55 1.66 79-90 e-0.8 0.43-0.48 (Fioravante, 2000)

SLB sand (sub-

round) 0.62 1.11 82-130

(2.17-e)2

1+e 0.44-0.53

(Hoque & Tatsuoka,

2004)

Ottawa sand

No. 20-30 0.72 1.20 69

(2.17-e)2

1+e 0.5 (Hardin & Richart, 1963)

Quiou car-

bonate sand 0.75 4.40 71 e-1.3 0.62



Maniscalco, 1993)


32

Another approach has been presented recently by (Wichtmann, Navarrete Hernández, &

Triantafyllidis, 2015) based on an extensive lab testing program and their earlier correlations

given for clean sands in (Wichtmann & Triantafyllidis, 2009). They introduced formulas de-

pending on CU and FC for the intrinsic parameters a, n and A for sands with non-cohesive fines.

They modified Equation 2-22 as:

Gmax = A(a – e)

2

(1 + e)p'

n p

atm 1 – n Equation 2-26

a = 1.94 exp(–0.066 CU) exp(0.065 FC) Equation 2-27

n = 0.40 CU 0.18[1+0.116 ln(1+FC)] Equation 2-28

A = (1563+3.13 CU 2.98) 0.5 [exp(–0.030 FC

1.10) + exp(–0.28 FC0.85)] Equation 2-29

These correlations can be used for FC < 10%. For FC > 10% CU should be taken as the

average inclination of the gradation curve above 0.063 mm and not as the regular d60/d10. It

should be noted, that their tested soils were obtained by mixing a natural quartz sand with quartz

powder, so the fines investigated were nonplastic. Also, if FC=0% is substituted into Equations

2-27 to 2-29, they lead back to their correlations given for clean sands in (Wichtmann &

Triantafyllidis, 2009).

The effect of plasticity of fines was studied more in detail in (Carraro, Prezzi, & Salgado,

2009). They investigated and compared the behavior of Ottawa sand mixed with plastic and

nonplastic fines to clean Ottawa sand. Concerning small strain stiffness, they made several im-

portant observations: large reductions were observed in Gmax for mixtures of sand with non-

plastic fines compared to clean sands; however, the addition of plastic fines up to 10% did not

result in a change at the same relative density and stress state. This can be explained by micro-

mechanical considerations according to the authors. They argued that nonplastic fines usually

do not have well-developed contacts with the sand particles and hence do not contribute to shear

stiffness, only reduce void ratio. For the range of fines content studied (15%), clayey fines

(plastic fines) are more compressible and get entrapped between sand contacts and they reshape

themselves during isotropic compression better than silt (nonplastic fines), which leads to a

fabric that transmits loads more effectively at very small strains. This shows the importance of

the role of soil fabric and structure on the mechanical behavior of soils.


33

Figure 2-16 Effect of plasticity on the small strain shear modulus of mixtures of Ottawa sand

with fines based on (Carraro, Prezzi, & Salgado, 2009). Note: Clayey sand is sand with plas-

tic fines; silty sand is sand with nonplastic fines

Although (Carraro, Prezzi, & Salgado, 2009) did not give a general equation for Gmax of

sands with plastic fines, their results for their Ottawa sand mixed with different % of Kaolin

clay (PI=26%) is shown here:

Table 2-2 Obtained intrinsic parameters for Ottawa sand with plastic fines based on

(Carraro, Prezzi, & Salgado, 2009)(using Equation 2-22 for Gmax in MPa with p’ in kPa)

FC of Ottawa sand

[%]

d50

[mm]

CU

[-]

A

[-]

F(e)

[-]

n

[-]

0 (clean) 0.31 1.89 61.1 (2.17-e)

2

1+e 0.437

2 - - 60.2 (2.17-e)

2

1+e 0.492

5 - - 56.9 (2.17-e)

2

1+e 0.480

10 - - 56.0 (2.17-e)

2

1+e 0.470


34

A different void ratio function was suggested by (Biarez & Hicher, 1994) for all soils

with liquid limit wl <50%, see Table 2-3.

Table 2-3 Correlation for Gmax of entire soil group given by (Biarez & Hicher, 1994)

Soil tested emin

[-]

emax

[-]

A

[-]

F(e)

[-]

n

[-]

All soils with

wl <50% 0.4 1.8 59

1

𝑒 0.5

Another correlation is often used because of its simplicity to estimate Gmax of coarse

grained soils which was introduced in (Seed & Idriss, 1970) and used later in (Seed, Wong,

Idriss, & Tokimatsu, 1986) (converted here to SI units):

Gmax=218.8 K2,max p0.5 Equation 2-30

with Gmax and p in [kPa] and a dimensionless coefficient K2,max. (Seed, Wong, Idriss, &

Tokimatsu, 1986) suggested K2,max values obtained from laboratory tests from 30 for loose sands

to about 75 to dense sands (Table 2-4) and values between 80-180 for gravelly soils.

Table 2-4 Estimation of K2,max for sands based on (Seed & Idriss, 1970)

Void ratio e [-] K2,max [-] Rel. density Dr

[%]

K2,max [-]

0.4 70 30 34

0.5 60 40 40

0.6 51 45 43

0.7 44 60 52

0.8 39 75 59

0.9 34 90 70

Based on resonant column tests on different soils and comparing results of a number of

researches (Santos & Correira, 2000) suggested a lower- and upper bound curve for sands and

clays which can be used as a check for the order-of-magnitude of the obtained Gmax value. The

curves are defined by:


35

Gmax = 4000 e–1.3 p' 0.5

Equation 2-31

Gmax = 8000 e–1.1 p' 0.5

Equation 2-32

for the lower-and upper bound respectively.

2.3.2 Modulus reduction

As with Gmax, the modulus reduction curve can also be estimated with correlations. Equa-

tion 2-25 can also be used to model the shear modulus degradation with increasing strain

(Jovicic & Coop, 1997). In this approach, coefficients A and n are dependent on shear strain

and thus cover a wide range of values. The parameter n is generally 0.5 for very small strains

and then increases toward 1.0 for large strains. This approach was used recently by (Oztoprak

& Bolton, 2013) who compiled a comprehensive laboratory test database for their state-of-the-

art paper with 454 tests gathered from 65 references. The database covers a wide variety of

granular soils, such as clean sands, gravels, sands with fines and/or gravels and gravels with

sand and fines, altogether 60 different soils. Based on this they developed new correlations for

strain dependent shear modulus and normalized degradation curves. They modified Equation

2-25 and suggested a different void ratio function for the strain dependent stiffness:

G(γ) = A(γ) patm

(1+e)3 (

p'

patm

) n(γ)

Equation 2-33

where A(g) and n(g) are strain dependent parameters, given by Table 2-5.

Table 2-5 Coefficients for Equation 2-33 in (Oztoprak & Bolton, 2013)

Shear strain amplitude g [%] A(g) n(g)

0.0001 5760 0.49

0.001 5520 0.51

0.01 4500 0.53

0.1 1810 0.73

1 370 0.86

Using shear strain amplitude of 0.0001%, Equation 2-33 can be used to estimate Gmax. An

important remark about the correlations given in (Oztoprak & Bolton, 2013) is that the accuracy

of the fit was also described in detail by the authors: Equation 2-33 with the suggested constants

for Gmax will provide estimates within a factor of 1.6 for one standard deviation of random error

(determined from 379 tests). This fairly large scatter suggests that it is extremely difficult to


36

give a general correlation for Gmax which would be reliable for such a wide variety of granular

soils.

Another approach to account for shear strain effects in laboratory testing, is to normalize

secant shear modulus Gsec(γ) as a function of shear strain by first dividing it by Gmax. If not

stated otherwise, the modulus G is Gsec in these normalized relationships. Then shear strain is

normalized by dividing it by a reference shear strain γref. Reference shear strain has an effect

similar to the confining stress correction. It may be obtained by two approaches:

γref

= γG/Gmax=Ratio

or γref

= τmax

Gmax Equation 2-34

where Ratio is some ratio less than 1.0, usually 0.7 or 0.5. This means the reference strain is

taken as a point of the degradation curve; corresponding to 70 or 50% degradation. This ap-

proach is used e.g. in the formulation of the Hardening Soil Small material model (Benz, 2006)

with 70% and in (Oztoprak & Bolton, 2013) with 50%. In the second approach, τmax is the

measured shear strength of the soil at a confining stress equal to that used in the test aiming at

measuring dynamic soil parameters. In this study the second approach was used. These normal-

izations are beneficial and frequently used by researchers because measured degradation curves

can be more easily compared to each other this way.

One of the earliest set of modulus degradation curves was published by (Seed & Idriss,

1970) to be used in ground response analysis. They obtained them from 75 tests on a total of

30 sands with a wide range of confinement pressure and void ratio. Together with the curves of

(Vucetic & Dobry, 1991) these are still frequently used because computer software such as

FLUSH (Lysmer, Udaka, Tsai, & Seed, 1975) and STRATA (Kottke, Wang, & Rathje, 2013)

offer them and they are easy to apply. The curves of (Vucetic & Dobry, 1991) were based on a

number of cyclic loading tests for a wide range of soils and the main focus was on the effect of

plasticity and over-consolidation, however the authors suggested a curve for coarse grained

soils (PI=0 curve) too. Both curves for sand are presented in Figure 2-17 and the original set of

curves by (Vucetic & Dobry, 1991) is given in Figure 2-18. While for clays the over-consoli-

dation ratio OCR has a significant effect on the shear modulus, for sands this is not the case and

the influence of OCR on Gmax is negligible (Jamiolkowski, Lancellotta, & Lo Presti, 1995).


37

Figure 2-17 Modulus reduction curves for sand

by (Seed & Idriss, 1970) and (Vucetic & Dobry, 1991)

Figure 2-18 Modulus degradation curves for clays and sand (PI=0)

by (Vucetic & Dobry, 1991)

Many authors also focus on the effect of cyclic degradation of stiffness; the decrease of

stiffness with cycle number can be seen in Figure 2-19. The higher number of cycles, the lower

G/Gmax value can be observed. This is due to contact deterioration, which decreases inter-parti-

cle friction.

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1 10

G/G

ma

x

Shear strain g (%)

G Sand S-I

G Clay PI=0 V-D


38

Figure 2-19 Effect of cyclic stiffness degradation on modulus reduction curves of clays and

sand (PI=0) by (Vucetic & Dobry, 1991)

While these degradation curves are useful, since they were not given with a formulation,

just as a set of numbers, their use is somewhat difficult, a more comprehensive mathematical

description allows for better modeling. For this, the most commonly used formula is the hyper-

bolic law for the normalized degradation curve, first proposed by (Hardin & Drnevich, 1972):

G

Gmax =

1

1+|γ

γref|

Equation 2-35

In this approach, gref is taken as tmax/Gmax. While the formulation was clear, the authors

observed, that this true hyperbolic relationship did not generally fit their data, so they introduced

a modification to the strain scale by using ‘hyperbolic shear strain’ which replaced g/gref on the

horizontal axis. For similar reasons, later (Darendeli, 2001) suggested a modified hyperbolic

model based on testing of intact sand and gravel samples:

G

Gmax=

1

1+(γ

γref)

a Equation 2-36

where exponent a is called curvature coefficient. It should not be confused with the coefficient

in the void ratio function in Equation 2-24 or Equations 2-26 and 2-27. Here gref is the shear

strain amplitude where the shear modulus reduces to 50%. To account for pressure dependence,

gref can be calculated as:


39

γref

= γref1

(p'

patm)

k Equation 2-37

where gref,1 is the reference shear strain at p’ = patm =100 kPa with an exponent k. In their detailed

study on the effect of the uniformity coefficient on G/Gmax, (Wichtmann & Triantafyllidis,

2013) suggested to use a=1.03, k=0.4 and

γref1

= 6.52 ∙ 10–4 exp[–0.59 ln (CU)] Equation 2-38

A similar formulation for the degradation curve was also used by (Oztoprak & Bolton, 2013).

They formulated their fit as a modified hyperbolic equation and provided mean, upper- and

lower bound curves based on the gathered vast quantity of earlier test data:

G

Gmax=

1

1+(γ – γeγref

)

a Equation 2-39

with the constants as

ge = 0%; gref = 0.02%; a =0.88 for the lower bound,

ge = 0.0007%; gref = 0.044%; a =0.88 for the mean,

ge = 0.003%; gref = 0.1%; a =0.88 for the upper bound,

where ge is the elastic threshold strain, beyond which the modulus drops below its maximum;

the reference shear strain gref is once again the strain at which G/Gmax=0.5; and a is the curvature

parameter. The additional strain parameterge allows covering effects of cementation and inter-

locking effects at small strains. Figure 2-20 shows the degradations curves and the constants

with slightly different notation.

If there is no further data available, the authors suggested calculating the elastic threshold

strain and the reference strain for the most common confining stresses (70 kPa – 600 kPa) as:

γe [%] = 8∙10

–5 (p'

pa) + 6∙10

–4 Equation 2-40

γref

[%] = 0.008 (p'

pa) + 0.032 Equation 2-41


40

Figure 2-20 Best fit modulus degradation curves for sands by (Oztoprak & Bolton, 2013)

To assess the effects of uniformity coefficient CU and void ratio e and relative density ID,

the authors performed a multivariable regression analysis and suggested the following, more

detailed equations:

γref

[%] = 0.01 CU–0.3 (

p'

pa) + 0.08 e ID Equation 2-42

γe [%] = 0.0002 + 0.012 γ

ref Equation 2-43

a = CU–0.075

Equation 2-44


41

According to Equation 2-42, an increase in CU leads to a decrease in gref which means a

swifter loss of elastic stiffness with strain. (Oztoprak & Bolton, 2013) argued that strain incom-

patibility between the fine matrix and the larger particles may be behind this effect; because the

presence of larger particles causes premature sliding of the smaller ones in contact with them.

If their contact is sliding, the small particles no longer contribute to global shear stiffness; hence

a greater disparity in grain size reduces gref which is also true for ge. Another interesting finding

is that gref correlates to the group e*ID rather than e. The authors’ discussion stated that e*ID

may be related to grain shape, since:

high void ratio for a rounded sand would indicate low relative density, so the product e*ID

would also be small

while an angular sand could have a high void ratio at a high relative density and give a

large product e*ID

The estimates based on Equations 2-37, 2-40 to 2-42 for shear modulus reduction G/Gmax

are accurate within a factor of 1.13 for one standard deviation of random error (from 3860 data

points). This suggests that while the estimation of Gmax with Equation 2-30 is quite unreliable,

the degradation curve can be estimated fairly well. As a conclusion, it is suggested to obtain

Gmax either with in-situ or with laboratory measurement methods if possible.

Application of correlations for Gmax to in-situ soils needs further considerations. It is ev-

ident from many laboratory experiments, that it is fairly difficult to reproduce in situ conditions;

particularly the effects of aging, moisture conditions and cementation may be the reason behind

in-situ small strain stiffness being usually higher than stiffness measured in the laboratory.

Aging causes a continuous increase in Gmax; according to observations an approximately

linear increase can be observed with the logarithm of time past the end of primary consolidation

which cannot be attributed solely to the effects of secondary compression according to (Kramer,

1995). The increase of stiffness with time is usually described by:

ΔGmax=NG(Gmax)1000 Equation 2-45

where Gmax is the increase in Gmax over one log cycle of time and (Gmax)1000 is the stiffness at

a time of 1000 minutes after the end of primary consolidation. NG is a material constant which

can be obtained in laboratory measurements. This effect is more pronounced for fine grained

soils. According to (Afifi & Richart, 1973) for a soil with d50 higher than 0.04 mm, the increase

is less than 3% per log cycle of time. (Wichtmann & Triantafyllidis, 2009) reported an NG value

of 0.5% for a quartz sand with d50 = 0.6 mm.


42

Moisture conditions can also change soil behavior and the effect of capillary pressure

should definitely be considered if the in-situ condition of the layer is unsaturated. Suction will

result in an increase in effective stress and this will cause an increase in Gmax compared to dry

and fully saturated soils. An estimated increase in p’ can be applied in the correlations to ac-

count for this effect. Detailed measurements on capillary effects performed on four sands and

a silt are presented in (Wu, Gray, & Richart, 1984), a similar study focused on the small strain

stiffness of unsaturated compacted soils presented in (Sawangsuriya, 2006).

Cementation is difficult to account for as undisturbed samples are usually unaffordable

and may not be representative for a larger soil mass while cementation is difficult to reproduce

on disturbed samples in the laboratory.

2.3.3 Damping

According to theory below the elastic threshold strain ge no hysteretic behavior occurs, so soil

response is linear and no damping is present. However, experimental evidence shows that some

energy is dissipated even at very low strain levels according to (Kramer, 1995). Above the

threshold strain, hysteretic behavior and hence damping increases with strain, as shown in Fig-

ure 2-7.

Estimating damping with correlations is perhaps a bigger challenge than estimating stiff-

ness, because measurement of damping is more sensitive and difficult. Although the measure-

ment of damping is not as precise as measurement of stiffness, in practical calculations this may

not be a big issue, as the effect of damping is minor in most calculations compared to the effect

of stiffness. (Hardin & Drnevich, 1972) and (Tatsuoka, Iwasaki, & Takagi, 1978) proposed to

express damping ratio as a function of G/Gmax:

D = f (G

Gmax) Equation 2-46

Such an approach was suggested by (Ishibashi & Zhang, 1993) who summarized many

earlier research and proposed the following equation for sands:


43

D = 0.333 {0.586 (G

Gmax)

2

– 1.547 (G

Gmax) +1} Equation 2-47

Later (Zhang, Andrus, & Juang, 2005) published a correlation based on RC and TOSS

testing that has also been used by many authors:

D = c1 (G

Gmax)

2

+ c2 (G

Gmax) – (c1+c2)+Dmin Equation 2-48

where c1 = 0.094 and c2 = –0.265 for RC; c1 = 0.106 and c2 = –0.316 for TOSS; and Dmin is the

minimum measured damping at very low strains. In their detailed study on the effect of the

uniformity coefficient on D, (Wichtmann & Triantafyllidis, 2013) suggested using:

c1 = 0.26 – 0.074 ln(CU) and c2 = –0.59 + 0.158 ln(CU) Equation 2-49

As with Gmax, the most commonly used correlations for ground response analysis were

published by (Seed & Idriss, 1970) and (Vucetic & Dobry, 1991), shown in Figure 2-21.

Figure 2-21 Damping curves for sand by (Seed & Idriss, 1970) and (Vucetic & Dobry, 1991)

(Vucetic & Dobry, 1991) provided a set of damping curves in their paper. Here again the

effect of OCR and PI was in focus, but the curve obtained for PI = 0 is widely used for sands

in ground response analysis and one can compare the behavior of sand to fine grained soils

easily based on Figure 2-22.

0

5

10

15

20

25

30

0.0001 0.001 0.01 0.1 1 10

Dam

pin

g r

atio

D[%

]

Shear strain g (%)

D Sand S-I

D Clay PI=0 V-D


44

Figure 2-22 Damping curves for clays and sand by (Vucetic & Dobry, 1991)

3 RC-TOSS testing

3.1 Basic testing concepts of RC and TOSS

For the Resonant Column (RC) tests, usually a cylindrical specimen is driven in a harmonic

manner back and forth in torsion at very low strain amplitudes with varying frequencies in order

to find resonance. Resonance frequency is found when the acceleration response amplitude is

reaching a maximum. The governing equation for this problem is based on the theory of wave

propagation in rods and can be written as:

II0

= ωnLvs

tanωnLvs

= β tan β Equation 3-1

where I is the mass polar moment of inertia of the sample; I0 is the free end mass polar moment

of inertia; n is the resonant frequency in torsion; L is the height of the sample; vs is the shear

wave velocity of the sample and is the specimen/device constant. All the variables in Equation

3-1 are known, except ωn and vs; so by the measurement of n shear wave velocity can be

obtained by:

vs = ωn L

β Equation 3-2

Small strain shear modulus is then calculated as:


45

Gmax = ρ vs2 Equation 3-3

where is the density of the sample.

The torsional loading is most often achieved with a drive head placed on the top of the

sample containing magnets which are surrounded by coils. The coils are driven by a sinewave

generator and amplifier while the frequency is varied until resonance is found by observing the

peak response from an accelerometer connected to the drive head. Displacement d can be com-

puted by integrating the acceleration twice. For the driving harmonic acceleration, peak dis-

placement dpeak is simply related to peak acceleration apeak as:

dpeak=apeak

ωn2 Equation 3-4

Damping can be obtained by the logarithmic decrement method and the half bandwidth

method in RC tests. The former one is done by releasing the sample by turning off the excitation

at resonant frequency. At higher amplitudes, it is often difficult to get stable readings for the

latter method so the log-decrement method generally works better.

The value of damping by the logarithmic decrement method is obtained by:

δ = 1N ln (

z1z1+N

) Equation 3-5

D = 1

√1+(2πδ

)2

≈ δ

2π Equation 3-6

where is log decrement, N is the number of cycles between two values, zN is amplitude at

cycle N, D is damping ratio. The simplified approximation is accurate to about D=0.30.

Torsional Simple Shear Testing (TOSS) involves a similar loading pattern as the RC test,

but at much lower frequencies. Rotation is measured using non-contacting proximitors and

torque is calibrated from the electrical current level through the coils. Stress and strain can be

computed based on sample geometry. Additionally, a vertical LVDT on the center axis of the

drive head can be used to measure changes in specimen height during testing. A general loading

usually consists of many harmonic loading cycles at certain amplitude to be able to assess the

effects of loading cycle numbers and reach a “steady state” hysteresis loop. The test can either

be stress controlled or strain controlled. Interpretation of results is based on evaluation of the

obtained hysteresis loops in the shear stress-shear strain diagram which is shown in Chapter

3.4.


46

Comparing RC to TOSS testing the following common features and drawbacks can be

highlighted. Benefit of the TOSS testing over RC testing is that non-harmonic, irregular load

histories can also be applied to the specimen. Any chosen cyclic loading can be considered

irregular history, e.g. an earthquake record or consecutive cyclic loading with gradually chang-

ing amplitude. RC testing is usually capable of obtaining shear stiffness and damping at the

very small strains (10-4%) and up to intermediate strains (10-2%). Due to measurement precision

TOSS testing is not capable to capture the behavior at very small strains but can be used up to

larger strains (10-3% - 10-1%) depending on device characteristics. In other words RC is best

for the very low strain range and precise measurement of Gmax, while TOSS can obtain most

part of the degradation curve. These two tests are usually performed in separate devices, how-

ever in this study a combined device was used which is capable of performing both tests on the

same specimen, as shown in Chapter 3.2.

3.2 Device

The combined Resonant Column-Torsional Simple Shear Device (RC-TOSS device) used in

this study was originally built by Prof. Richard Ray at Michigan University in the 1980s (Ray

R. P., 1983), (Ray & Woods, 1988). For this research, the device has been rebuilt and calibrated

in Győr at the Geotechnical Laboratory of the Department of Structural and Geotechnical En-

gineering at Széchenyi István University and it is now capable of loading and measuring at

higher torques and wider strain ranges. The testing system has been improved with respect to

testing range, data storage and manipulation. MS Excel is used for managing testing, VBA

macros have been developed for handling TOSS cycle plotting, TOSS and RC data logging as

well as comparing and presenting them, see Annex B. Resonant column testing is regulated by

the testing standard (ASTM, 2015); however, the scope of the standard does not cover hollow

cylindrical testing or combined RC-TOSS testing. Therefore, a detailed testing procedure in-

cluding device assembly and sample preparation is presented in Chapter 3.3 and in Annex A.

Figure 3-1 shows the hollow cylinder sample inside the black membrane with the drive head

on top, surrounded by the four drive coils.


47

Figure 3-1 Combined Resonant Column – Torsional Shear Device (RC-TOSS)

at the Geotechnical Laboratory of Széchenyi István University

The device can drive a cylindrical specimen (inner diameter Di = 4.0 cm, outer diameter

Do = 6.0 cm, length L = 14 cm) in torsion with the use of permanent magnets and electric coils.

The soil specimen is covered by latex membranes inside and out and is fixed at the base and

free to rotate at the top, where the drive head is attached, in other words it is a fixed-free con-

figuration. The drive head consists of a coil and magnet system where there is open space be-

tween the magnet and coils. The cross-section of the device can be seen in Figure 3-2.

An accelerometer is used for measuring displacement during RC testing; the nature of

acceleration measurement allows for very high accuracy at low amplitudes (γ=10-4 %). Prox-

imitors measure the gap distance between their targets and are used for measuring rotation

(strain) during TOSS tests. Since they are DC output devices, there is no low-frequency roll-

off as in accelerometer measurements. A linear variable differential transformer (LVDT)

measures vertical displacement of the drive head. Its inner core is connected by a rigid threaded

rod to the drive head. The other end of the rod is suspended on a spring, providing a counter-

force to the drive head’s weight without impeding the RC measurements.

Coil

Magnet

LVDT

Proximitor

target

Accelerometer

Sample


48

The upper and lower specimen rings have a porous interface and can provide saturation

to the specimen. Pore pressures are measured at the confining cell’s outer port. Due to the ex-

tensive electronics, the confining medium is air.

Figure 3-2 Cross section of RC-TOSS testing device

The driving system allows for individual control of the applied shear stresses as well as

the cell pressure (with use of a pressure chamber) and the hollow cylindrical specimens have

Measurement

post

Proximitor:

Mount

Target

Top:

Retainer

Plug

Ring

Drive coil:

Mount

Support

Base:

Plug

Ring

Drive head

Outer mold

Membrane

Soil specimen

Base plate

Inner mold

Pore water

Pore stone

Magnet


49

the great advantage of having nearly-uniform shear stress distribution throughout its cross-sec-

tion. Calibration methods are described in detail in (Ray R. P., 1983).

For this testing system, typical resonance is around 300 rad/sec which means if the peak

acceleration signal is 5mV, acceleration is 0.015 m/sec2, displacement is 1.67x10-7m, and peak

strain around 10-4%.

Obtaining damping with the RC method requires capturing to free decay of the sample

after being driven at resonance frequency. By using a cut-off switch that turns the coils into

open circuits there is very little coil-generated damping during log decrement measurements.

The sample will continue to resonate and response amplitude will be decreasing monotonically

due to energy dissipation between particles. The simple open circuit generated no counter-emf

damping when tested with a calibration rod, i.e. the rod decayed exactly the same amount, with

or without the magnet-coil assembly (with the open switch) in place.

Additionally to the regular RC testing, the drive system can be given an offset DC voltage

to apply static stress/strain then resonated with a low-amplitude signal to examine effects of

different stress ratios (1/3) and static shear stress/strain (tstatic,gstatic) on shear modulus, how-

ever the scope of this study only contained isotropic testing.

For TOSS testing with use of the data acquisition system and computer, various cyclic

and arbitrary load histories can be applied. Hysteresis and secant/tangent moduli can then be

computed from the acquired data. The unique design of the testing device allows for both RC

and TOSS tests to be run on the same specimen any number of times and in any sequence. A

particularly useful sequence is to perform low-amplitude RC between different stages of TOSS

testing to evaluate the effects of each TOSS stage on the “fundamental” property Gmax. The

baseline value of Gmax allows data to be normalized by reference strain (gref) or other quantities.

The drive system uses a 500-watt DC power amplifier that can be current-controlled. Under

current (rather than voltage) control, the device can consistently deliver the same stress level

during a test, even if the coils heat up. Using neodymium magnets, the device can generate

shearing stresses greater than t = 200 kPa.

3.3 Sample preparation

Two sample preparation methods were used in this study for the RC-TOSS test: dry filling with

tapping, and pluviation. A loose state was achieved by pouring dry soil into the sample mold

with a glass funnel. Since the hollow cylinder specimen had a wall thickness of 1 cm, a funnel

with a 14 cm long spout was used. During pouring, the end of the spout was continuously kept


50

near the surface of the soil and gently moved along, creating a bulking effect. This method is

similar to that used to measure the minimum density state. After pouring in the sample, the

mold was lightly tapped 10-15 times on its side to achieve a slightly denser state. The dense

condition was created by dry pluviation from a height of 50 cm. The soil was carefully poured,

so that the thread of soil would be the thinnest achievable. For the tested sands, a separate

comparison indicated that dry pluviation and vibration produced nearly identical densities. Ta-

ble 4-1 shows void ratios achieved by dry filling and dry pluviation methods.

The inner and outer mold held the respective membranes during sample preparation. Once

the soil was placed, both membranes were carefully pulled over to the top ring and sealed in

position by O-rings. An 80 kPa vacuum confinement was activated inside the sample, the inner

and outer molds were removed and the drive system and transducers were assembled. Specimen

diameter and height were also measured. A detailed, step-by-step summary about device as-

sembly and sample preparation is given in Annex A.

A typical test started with low amplitude RC measurements after 15 min. confinement.

Effect of confinement time was not investigated in detail for this study; only for one sample

since granular material under dry conditions do not produce any appreciable duration of con-

finement effects. For higher confinement stresses a pressure chamber was placed over the test-

ing device and regulated air pressure was applied while keeping the vacuum confinement on.

At the next test stage, high amplitude RC and a series of TOSS tests were performed with

gradually increasing strain levels. The TOSS tests were stress controlled and each stage con-

sisted of 100 loading cycles. Finally, a low amplitude RC measurement was performed to check

if TOSS loading resulted in a change of small strain stiffness.

3.4 Interpretation of measurement results

The measurement with the combined RC-TOSS device is governed with a PC using a

data acquisition system and MS Excel with VBA codes. A detailed description of the VBA

codes used for running and logging the tests can be found in (Ray R. P., 1983). RC testing is

controlled by manipulating the function generator with feedback about acceleration response

provided by an oscilloscope. Both tests are logged and recorded into a single macro enabled

spreadsheet file (xlsm format) and further data interpretation is used to obtain figures such as

shear stress-shear strain curves or modulus reduction curves. Since the combined test could

consist of RC and TOSS measurements at several confinement stages with several shear stress

amplitudes at each stage with each one containing usually 100 cycles of TOSS loading, the


51

storage of raw test data usually results in a quite large xlsm file (10-20 MB). This is the reason

why data interpretation was chosen to be done in a separate macro enabled spreadsheet file with

several macros developed for this study to produce material parameters and graphs.

Data interpretation generally starts with the selection and copying of all performed RC

test sequences and the selection and copying of recorded shear stress and shear strain data of

TOSS cycles to the prepared spreadsheet used for interpretation. Interpretation of RC measure-

ment data is fairly easy, because measured values of resonance period and acceleration Volts

(RMS) are used to calculate shear wave velocity and shear modulus already in the testing

spreadsheet; and the logarithmic decrement method is also coded into it, so damping values are

readily shown in the testing spreadsheet, see Figure 3-3. TOSS interpretation needs some fur-

ther considerations. During the TOSS test for each cycle the readings of the DA system are all

stored (vertical LVDT reading, pore pressure reading, proximitor readings, stress reading and

calculated strain). It is important to keep and store these data in case spurious results are found

and further analysis is required on the raw data.

After choosing and copying the raw data, several subroutines and functions (HystSeg-

ment2, CalcStrainSA, CalcSecantMod, DampSec, see Annex B) developed for this study in

VBA are used to obtain measured material parameters. These include the calculation of the area

of a hysteresis loop with a numerical integral approach, calculation of single amplitude strain,

secant shear modulus and damping based on a set of stress-strain values of a single loop meas-

ured during TOSS testing. RC testing data is stored on a separate tab (RCTestSheet); each

stress-strain value of a TOSS cycle are placed in the A and B columns of separate tabs. Then a

macro (Grabdamp, see Annex B) is used to gather all calculated material parameters to a sum-

mary tab (Results), this is shown in Figure 3-4.

Finally stress-strain curves, modulus reduction curves and damping curves can be gener-

ated based on the interpreted data. Typical results of RC testing will be shown in detail in Chap-

ters 4.2 and 4.4.

Typical results of a single TOSS test with 100 cycles are shown in Figure 3-5. Two main

effects can be observed. With increasing number of cycles the hysteresis loop gets thinner and

the secant of the loop gets steeper. This means that cyclic degradation of damping and cyclic

stiffening can be observed for this material. These effects have also been found by (Kramer,

1995) for drained cyclic testing of sands. Note that hysteresis loops of cycles 75 and 100 are

exactly on top of each other. This means these two cyclic effects disappear after a number of

cycles which was around 75 for this soil.


52

Figure 3-3 Raw RC testing data in the measurement controlling spreadsheet

Figure 3-4 Interpreted TOSS measurement data. Two confinement stages and several shear

stress amplitudes are shown with cycle 1 and cycle 100

Date/Dátum Sample/Minta Inside

Dia/

Belső

átm. (cm)

Outside

Dia/

külső

átm. (cm)

Length/

Magass

ág (cm)

Sample

Volume/

Térfogat

(cm3)

Wet Wt/

Nedves

tömeg (g)

Dry Wt./

Száraz

tömeg (g)

Water

content/

Víztartalo

m (%)

Wet Unit

wt/

Nedves

térfogatsű

r. (g/cm3)

Dry Wt/

Száraz

térfogatsűr

(g/cm3)

Void

ratio/

Hézagtény

ező (-)

2015.12.02 (34)P2 4.01 5.94 13.82 208.59 394.51 381.58 3.39 1.891 1.829 0.454

Date Time/Vizsgálat

ideje

Elapsed/

Eltelt idő

Period/

Periódusi

dő

Freq./

Frekven

cia

Shear wave

velocity/

Nyíróhullám

seb. Vs

Shear

modulus/

Nyírási

modulus

Accel

Volts/

Elmozdul

ás

Accel

Disp/

Elmozdul

ás (fél

Strain/

Fajl.

alakváltoz

ás

G/Gmax gam/gamref Z Z+1 Dampi

ng/

Csillap

ításConf Pres (kPa) (clock) (hr:min:sec) (msec) (rad/sec) (m/sec) (kPa) (volts rms) (cm s-a) (% s-a) (%)

84 2015. 11. 30. 17:50 1:35:00 14.76 425.69 243.81 108721 0.00703 1.80E-04 7.04E-04 100% 2.04E-02Taumax (kPa) 2015. 11. 30. 17:59 1:44:00 14.77 425.40 243.64 108573 0.00662 1.70E-04 6.64E-04 100% 1.93E-02

38 2015. 11. 30. 19:20 3:05:00 14.76 425.69 243.81 108721 0.00695 1.78E-04 6.96E-04 100% 2.02E-02Gmax (kPa) 2015. 12. 1. 9:05 16:50:00 14.74 426.27 244.14 109016 0.00733 1.87E-04 7.32E-04 100% 2.12E-02

109016 2015. 12. 1. 9:10 16:55:00 14.96 420.00 240.55 105833 0.02133 5.61E-04 2.19E-03 97% 6.36E-02 30 24 0.71Gam Ref (%) 2015. 12. 1. 9:15 17:00:00 15.10 416.10 238.32 103880 0.03690 9.88E-04 3.87E-03 95% 1.12E-01 22 17 0.82

3.45E-04 2015. 12. 1. 9:20 17:05:00 15.29 410.85 235.31 101274 0.05530 1.52E-03 5.94E-03 93% 1.72E-01 29 18 1.522015. 12. 1. 9:25 17:10:00 15.38 408.53 233.98 100132 0.06860 1.91E-03 7.46E-03 92% 2.16E-012015. 12. 1. 9:30 17:15:00 15.85 396.42 227.04 94281 0.08800 2.60E-03 1.02E-02 86% 2.95E-01 22 14 1.552015. 12. 1. 9:35 17:20:00 15.91 394.92 226.19 93572 0.11570 3.44E-03 1.35E-02 86% 3.90E-01 29 16 1.99

2015. 12. 1. 10:42 18:27:00 16.16 388.81 222.69 90699 0.14300 4.39E-03 1.72E-02 83% 4.98E-01 30 15 2.212015. 12. 1. 10:42 18:27:00 16.71 376.01 215.36 84827 0.17660 5.79E-03 2.27E-02 78% 6.57E-01

Conf Pres (kPa) 2015. 12. 1. 17:26 25:11:55 12.58 499.46 286.06 149666 0.00510 9.48E-05 3.71E-04 99% 7.82E-03160 2015. 12. 1. 17:31 25:16:20 12.53 501.45 287.20 150863 0.00508 9.37E-05 3.67E-04 100% 7.73E-03

Taumax (kPa) 2015. 12. 1. 18:29 26:14:23 12.54 501.05 286.97 150622 0.00500 9.24E-05 3.61E-04 100% 7.62E-0372 2015. 12. 1. 18:31 26:16:23 12.54 501.05 286.97 150622 0.00440 8.13E-05 3.18E-04 100% 6.70E-03

Gmax (kPa) 2015. 12. 1. 18:29 26:14:23 12.53 501.45 287.20 150863 0.00893 1.65E-04 6.44E-04 100% 1.36E-02150863 2015. 12. 1. 18:29 26:14:23 12.55 500.65 286.74 150382 0.01295 2.40E-04 9.37E-04 100% 1.98E-02 35 29 0.60

Gam Ref (%) 2015. 12. 1. 18:29 26:14:23 12.55 500.57 286.70 150335 0.02390 4.42E-04 1.73E-03 100% 3.65E-02 29 22 0.954.74E-04 2015. 12. 1. 18:29 26:14:23 12.55 500.65 286.74 150382 0.02780 5.14E-04 2.01E-03 100% 4.24E-02

2015. 12. 1. 18:29 26:14:23 12.69 495.13 283.58 147083 0.04520 8.55E-04 3.35E-03 97% 7.05E-02 30 23 0.922015. 12. 1. 18:29 26:14:23 12.80 490.87 281.14 144565 0.06940 1.34E-03 5.23E-03 96% 1.10E-01 16 8 1.102015. 12. 1. 18:29 26:14:23 12.88 487.82 279.40 142775 0.09760 1.90E-03 7.44E-03 95% 1.57E-01 24 11 1.212015. 12. 1. 18:29 26:14:23 12.90 487.07 278.96 142333 0.12500 2.44E-03 9.56E-03 94% 2.01E-01 28 14 1.162015. 12. 1. 18:29 26:14:23 13.12 478.83 274.24 137558 0.17940 3.63E-03 1.42E-02 91% 2.99E-01 22 9 1.422015. 12. 1. 18:29 26:14:23 13.25 474.20 271.59 134913 0.20440 4.22E-03 1.65E-02 89% 3.48E-01 26 10 1.54

TOSS test/TOSS vizsgálatStrain/Fajlagos alakváltozás

fél amplitúdója (s-a %)

Gsec

(kPa)

Damping/

Csillapítás

(-)

Csillapítás/

Damping

(%)

Gsec/Gmax (-) g/gref (-)Damping

RO (%)

Damping

Ishibashi

1993 (%)

Damping

Zhang 2005

(%)

(34)P2 @84kPa

cy1_CyTest 14_5kPa 0.00667 75018 0.04 4.10

cy100_CyTest 14_5kPa 0.00662 75504 0.01 1.34 0.69 0.19 2.59 6.98 4.91

cy1_CyTest 17_10kPa 0.01296 77194 0.02 1.93

cy100_CyTest 17_10kPa 0.01288 78157 0.02 1.57 0.72 0.37 4.43 6.40 4.50

cy1_CyTest 19_20kPa 0.02690 74323 0.04 4.15

cy100_CyTest 19_20kPa 0.02680 74583 0.03 3.25 0.68 0.78 7.14 7.19 5.05

cy1_CyTest 20_30kPa 0.05024 59688 0.11 10.93

cy100_CyTest 20_30kPa 0.04560 65768 0.06 5.61 0.60 1.32 9.35 9.32 6.50

cy1_CyTest 22_40kPa 0.08495 47142 0.16 15.63

cy100_CyTest 22_40kPa 0.06981 57226 0.08 8.43 0.52 2.03 11.12 11.64 8.04

cy1_CyTest 23_50kPa 0.14750 33874 0.21 20.92

cy100_CyTest 23_50kPa 0.10393 48049 0.12 11.68 0.44 3.01 12.71 14.39 9.84

(34)P2 @160kPa

cy1_CyTest 25_5kPa 0.00366 136598

cy100_CyTest 25_5kPa 0.00370 134856

cy1_CyTest 26_10kPa 0.00755 132488 0.01 0.83

cy100_CyTest 26_10kPa 0.00748 133558 0.01 0.72 0.89 0.16 2.17 2.99 1.93

cy1_CyTest 28_20kPa 0.01536 130108 0.01 1.32

cy100_CyTest 28_20kPa 0.01542 129483 0.01 1.19 0.86 0.33 3.99 3.46 2.29

cy1_CyTest 30_30kPa 0.02494 120171 0.04 3.75

cy100_CyTest 30_30kPa 0.02431 123226 0.02 2.25 0.82 0.51 5.53 4.24 2.86

cy1_CyTest 31_40kPa 0.03513 113801 0.05 4.90

cy100_CyTest 31_40kPa 0.03425 116608 0.03 3.09 0.77 0.72 6.84 5.14 3.51

cy1_CyTest 33_50kPa 0.04624 108045 0.06 5.72

cy100_CyTest 33_50kPa 0.04535 110089 0.04 3.96 0.73 0.96 7.99 6.10 4.18

cy1_CyTest 35_60kPa 0.05897 101655 0.07 6.62

cy100_CyTest 35_60kPa 0.05808 103134 0.05 4.95 0.68 1.22 9.02 7.20 4.95

cy1_CyTest 36_70kPa 0.09049 77296 0.11 11.46

cy100_CyTest 36_70kPa 0.08095 86328 0.07 7.03 0.57 1.71 10.42 10.21 6.99

cy1_CyTest 38_80kPa 0.10410 76776 0.10 10.33

cy100_CyTest 38_80kPa 0.09938 80358 0.08 8.36 0.53 2.09 11.26 11.40 7.77

cy1_CyTest 40_90kPa 0.13083 68720 0.13 13.02

cy100_CyTest 40_90kPa 0.12098 74263 0.10 9.97 0.49 2.55 12.05 12.67 8.61

cy1_CyTest 41_100kPa 0.16597 60183 0.16 16.05

cy100_CyTest 41_100kPa 0.14610 68322 0.12 11.83 0.45 3.08 12.79 13.97 9.46


53

In all further graphs, stiffness and damping values of cycle 100 was plotted for each TOSS

test unless noted otherwise. The magnitude of the effect can be assessed for each TOSS test,

because in the measurement results both cycles 1 and 100 are shown; see Annex C.

Figure 3-5 Effect of number of cycles in TOSS test

Stress-strain response to an irregular load history is shown in Figure 3-6. Note the inter-

mediate turnaround points resulting in smaller unload-reload loops with steeper secant value

showing small strain stiffness. Such load histories can be used to study whether the extended

Masing rules apply for the tested material, however the scope of this thesis did not comprise of

the detailed investigation of irregular load histories.

-100

-75

-50

-25

0

25

50

75

100

-0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100

Shea

r st

ress

t[k

Pa]

Shear strain g [%]

Cycle 0

Cycle 1

Cycle 5

Cycle 10

Cycle 20

Cycle 50

Cycle 75

Cycle 100


54

Figure 3-6 Stress-strain response to an irregular load history in TOSS test

Another topic of result interpretation has to be addressed concerning damping in the Ram-

berg-Osgood material model presented earlier. The formulation presented in Chapter 2.1.4 does

not give an equation for calculating damping directly. Therefore, a numerical integration pro-

cedure has been developed to calculate damping based on the hysteresis loop using Equation 2-

8. The integration of half of the loop can be performed along the stress axis with use of the

differential element shown in Figure 3-7.

Figure 3-7 Numerical integration for obtaining hysteretic damping with the Ramberg-Osgood

material model

dτ

f1(τi+1)

f1(τ

i)

f2(τi)

f2(τ

i+1)

γ1,τ1

γ

τ

γ2,τ

2


55

The formulation for f1 and f2 is the following:

f1(τ) = γ =

τ – τ1Gmax

(1+α |τ – τ1

2 C τmax|R–1

) + γ1 Equation 3-7

f2(τ) =

(τ – τ1)(γ2 – γ1)

τ2 – τ1 + γ

1 Equation 3-8

Let us introduce the variables a, b, c as:

a = 1

Gmax, b =

α

(2 C τmax)R–1 , c =

γ2 – γ1τ2 – τ1

Equation 3-9

This way f1 and f2 can be calculated as:

f1(τ) = a (τ – τ1)+ab (τ – τ1)R+ γ

1 Equation 3-10

f2(τ) = c (τ – τ1) + γ

1 Equation 3-11

Then the integration can be written as:

∫ f2(τ) –

τ2

τ1 f

1(τ) dτ = ∫(c – a)(τ – τ1) dτ – ∫ ab (τ – τ1)

R dτ Equation 3-12

The integration of the total loop, which is the dissipated energy WD then calculates as:

WD = 2 ((c – a)(τ2 – τ1)

2

2 –

ab(τ2 – τ1)R+1

R+1) Equation 3-13

Damping then can be calculated using Equation 2-8. If the loading results in a symmetric

loop, t1=-t2, the calculation is even easier. Note, that to calculate damping with a given g value,

t has to be first calculated with the Newton Raphson method, because Equation 2-11 cannot

be inverted. This calculation was programed into a VBA function (ROHystDampingNew) to

obtain damping with the Ramberg-Osgood model to be able to compare it with measured val-

ues. The code is presented in Annex B.

3.5 Application of measurement results in numerical modeling

This chapter focuses on the interpretation of measurement results needed to obtain material

parameters to be applied in numerical modeling. For this the Ramberg-Osgood material model

was chosen, which has been used by several authors recently for performing nonlinear ground

response analysis. The model has been implemented by Midas into their finite element code

(MIDAS Information Technology Co., 2014). To assess the modeling capabilities of the pro-

gram and to present how small strain stiffness can be accounted for in numerical modeling, a


56

Torsional Shear test was modeled in 3D in Midas GTS NX. This modeling task has never been

performed with a nonlinear material model capable of modeling modulus degradation.

Midas GTS NX is a comprehensive 3D geotechnical finite element software package. It

is used for modeling deep foundations, excavations, tunnels, embankments etc. Static as well

as dynamic calculations can be performed; consolidation, seepage and slope stability by the

strength reduction method can also be modeled. It is a well-known program used worldwide

with capabilities of modeling a wide range of geotechnical structures and problems.

When using geotechnical finite element programs, the processes of verification and vali-

dation are crucial, as discussed by (Brinkgreve & Engin, 2013). As they describe it, verification

is the process of showing that a model or method has been properly implemented in a computer

program; while validation is the process to make plausible that a computer model possesses the

essential features to analyze a real problem and that the results obtained with it are representa-

tive for the situation. The former is usually done by the software developer and the latter should

be done by the user when creating a model.

The test chosen for modeling provided the results shown in Figure 3-8 and Figure 3-9.

Figure 3-8 Modulus reduction curve obtained with Resonant Column-Torsional Simple Shear

device


57

Figure 3-9 Hysteresis loops of Torsional Shear tests at different stress (strain) levels

Finite element modeling of such tests is a challenge. The geometry of the test is funda-

mentally axisymmetric; however, since the load is torsional, it is out-of-plane and requires a

true 3D model using cylindrical coordinate system. A so-called 2.5D model or extended 2D

axisymmetric model cannot reproduce a torsional load perpendicular to the model plane.

In Midas GTS NX, axisymmetric 3D modeling can be performed by defining a global

cylindrical coordinate system or using the cylindrical element coordinate system. For this study

the latter was used. Figure 3-10 shows a part of the model with the finite element mesh. Hexa-

hedral high order elements with 20 nodes were used with an average size of 4x4x3 mm. Nodes

on the grey soil elements can be seen in blue in Figure 3-10. This mesh was chosen so that three

layers of elements would build up the cross section and so strain distribution could easily be

assessed within the 1 cm thick hollow cylinder.

Figure 3-10 Detail of finite element mesh of soil sample and steel ring (partially invisible)


58

Figure 3-10 also shows yellow elements on top of the hollow cylinder soil sample, repre-

senting the steel ring, which transfers the load to the soil sample. A part of this ring is switched

off to better show soil elements. The imposed torsional load is modeled as surface stress acting

in the rotational (θ) direction of the cylindrical element coordinate system. Without this option,

an imposed node load would act in a tangential direction, which does not reflect true behavior

in this case.

The mesh consisted of 4032 elements. Mesh size dependency was checked and even a

coarser mesh with one layer of high order elements and a total number of 162 elements showed

identical results.

Pinned boundary conditions were used on the bottom surface of the soil sample since in

the test the soil directly connects to the rough surface of the porous stone. It should be noted

that some stress irregularities were observed even in the fine mesh close to the base fixity when

using fully fixed boundary conditions (fixed against translation and rotation in each direction).

The irregularities with pinned fixities were minor, see Figure 3-11. Since the testing device is

free on top and the self-weight of the drive head is counterbalanced by a calibrated vertical

spring, no other fixities have to be introduced in the model.

Calculation stages consisted of initial stress generation, confinement activation (with free

face surface pressure acting in normal direction of the element faces) and torsional loading

stages. This can be done in the software by defining construction stage sets and setting up con-

struction stage analysis cases. An important modeling step is to set up the output control before

starting the calculation. Here parts of the mesh can be chosen which will be used for documen-

tation. While the calculation will be performed for the whole mesh, results will only be saved

for these selected elements, therefore result size can be reduced. This still might be crucial for

3D FEM calculations, e.g. one stage with the fine mesh produces around 800 MB results. Result

types can also be adjusted here and it should be remarked that strains in solid elements are

turned off by default which can cause some initial confusion for some users.

Formulation of the material model has been presented in Chapter 2.1.4. An additional

modeling detail is that an option of using the model in “shear only” mode is available in the

program, which was used in this study. Unfortunately, the effects of this option are not detailed

in the manual, however the online release notes explain that if this mode is used, shear modulus

will be different in each direction separately (Gxy, Gyz, Gzx), otherwise an equivalent shear mod-

ulus will be used (Geq). While this information is somewhat helpful, there seems to be no way

of inputting separate shear moduli and the formulation of Geq is also not part of the manual as

of the time of modeling.


59

Obtaining parameters for FEM models from various laboratory and field tests is also a

challenging task because it involves the solution of an inverse problem. Some geotechnical

FEM software offer built-in program modules for fitting curves of commonly used laboratory

tests, e.g. Soil Test module in Plaxis, so that the user can assess the effects of changing a single

parameter and chose the most appropriate set of parameters for the specific project. However,

laboratory tests such as the Resonant Column or Torsional Shear Test are not yet implemented

in any of the mentioned modules.

For this study, in order to obtain model parameters, the Gmax value measured in a low

strain Resonant Column test and the stress-strain values of a higher strain single hysteresis loop

obtained from the Torsional Shear test were used. The measurement results were imported into

MS Excel and the formulation of the material model was implemented into a Visual Basic code.

Then, with a set of initial values for gref and hmax the response of the model was calculated. Due

to the properties of the formulation, if a full hysteresis loop is used, the turnaround point has to

be specified in advance as well. Then, at each data point, the square of the error could be ob-

tained and sum of the squared errors could be calculated. In MS Excel, the Solver module was

then used to choose model parameters with the best fit, namely the nonlinear GRG method

(Generalized Reduced Gradient) was applied to minimize the sum of square of the errors by

allowing Solver to change the two model parameters and solve the inverse problem. The ob-

tained model parameters are summarized in Table 3-1.

Table 3-1 Ramberg-Osgood material model parameters obtained from back analysis

Parameter Value Dimen-

sion

Small strain stiffness from RC test Gmax 91 161 [kPa]

Reference shear strain γr 8.1372×10-4 [-]

Max. damping constant hmax 0.19043 [-]

Dry density γd 16.15 [kN/m3]

Poisson’s ratio υ 0.3 [-]

A practical remark has to be mentioned here; the GRG method works dependably if the

values of the model parameters are fitted are of a similar order of magnitude. This can be

achieved by simply scaling the parameters in the formulation (e.g. using a parameter value 100x

larger, and then dividing it by 100 in the formula).


60

Horizontal stresses after the 84kPa confinement step are shown in Figure 3-11. Some

minor irregularities can be observed near the base fixities.

Figure 3-11 Distribution of horizontal stresses after confinement activation

A one-way loading stress-strain curve is shown in Figure 3-13. Stress and strain values

were taken as the average of the values from the three layers of elements in the cross section.

This average was found to be almost identical to the value in the middle element, see Figure

3-12. Response of the inside element and outside element follow the same stress strain curve.

Figure 3-12 Comparison of shear stresses in cross section

0

20

40

60

80

100

120

0 0.001 0.002 0.003 0.004 0.005 0.006

Sh

ear

stre

ss t

[kP

a]

Shear strain g [-]

Average

Inside element

Outside element


61

Agreement between the test, the formulation and the FEM calculation is excellent as

shown in Figure 3-13.

Figure 3-13 Stress-strain curve by FEM calculation and MIDAS Ramberg-Osgood

formulation

Figure 3-14 and Figure 3-15 show the deformed mesh and the distribution of total defor-

mations in the mesh. The horizontal shear of the elements is clearly observable.

Figure 3-14 Deformed mesh at turnaround point of loading

0

2

4

6

8

10

0.0 0.00005 0.0001 0.00015

Shea

r st

ress

t[k

Pa]

Shear strain g [-]

Results obtained withFEM calculation

Ramberg-Osgoodformulation

TOSS test


62

Figure 3-15 Distribution of total displacements at turnaround point of loading

Figure 3-16 shows shear strain distribution in the sample, which is even throughout the

height of the sample. Also, the benefit of using a hollow cylinder for testing is demonstrated

here; strain difference within the sample would be considerably higher in a full cylinder.

Figure 3-16 Shear strains at turnaround point of loading


63

The calculations were performed at several shear strain levels and based on the obtained

shear stress and strain values; the secant shear modulus was calculated. Two-way loading was

also applied. For this, manual load stepping was found to be beneficial. Load step size around

turnaround points of the hysteresis loops should be gradually smaller to achieve a stable calcu-

lation. The developed manual load stepping method is shown in Figure 3-17.

Figure 3-17 Manual load stepping used for stable calculation of hysteresis loop

Obtained hysteresis loop overlaps with the test results perfectly. Result of a single loop is

shown in Figure 3-18.

Figure 3-18 Hysteresis loop obtained with FEM calculation

0.0010.002

0.0040.007

0.010.02

0.040.07

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.90.93

0.960.98

0.990.993

0.9960.998

0.999

1

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Lo

ad m

ult

ipli

er [

-]

Load steps [-]

-25

0

25

-0.0004 0.0000 0.0004

Sh

ear

stre

ss t

[kP

a]

Shear strain g [-]


64

Hysteresis loops for several stress levels are shown in Figure 3-19.

Figure 3-19 Hysteresis loops obtained from two-way sequential loading in Midas

Figure 3-20 shows the normalized modulus degradation curve compared to the test data.

Figure 3-20 Modulus reduction curve by FEM calculation and RC-TOSS tests

This study verified that the Ramberg-Osgood material model implemented in MIDAS

GTS NX v2014 is capable of modeling this static 3D cylindrical laboratory test with represen-

-40

0

40

-0.0012 0.0 0.0012

Sh

ear

stre

ss t

[kP

a]

Shear strain g [-]


65

tation of nonlinear material behavior. The model also captured modulus degradation and hys-

teretic behavior. Sequential two-way loading modeling showed very good agreement with test

results.

Some minor modeling difficulties have been pointed out. The use of manual load stepping

and smaller load step size before and after the turnaround point was found to be beneficial to

achieve a stable calculation. This confirms the general expectation, that to calculate more com-

plicated load histories, such as recorded earthquake load histories, a smaller load step size may

be required when using automated load stepping.

4 Testing program and results

4.1 Materials investigated and test matrix

Nine samples of granular soils were selected for testing in the laboratory program. They are all

fluvial sediments of the Danube River. Danube sands are present throughout Hungary and are

often encountered in large construction projects near the river as well as former river meanders.

They often have a wide variation of in-situ density and may contain a significant percentage of

fines; generally, with low plasticity. Other zones contain a significant percentage of gravel

which is very difficult to evaluate in laboratory tests. The samples were retrieved from depths

of 5-15 m from a hollow-stem auger sampler and weighed 2-3 kg. Main parameters of the tested

samples are summarized in Table 4-1. These samples are considered to be typical Danube sands.

Samples P1-P7 originate from the direct vicinity of the present channel of the River Dan-

ube around Paks. The stratigraphy of the coastline consists of Holocene age fluvial sediments;

however, this fine sand terrace material is often mixed with aeolian particles. Original surfaces

were probably deflated and accumulated in an alternating fashion. This diverse formation

shaped the grain structure to its current state. Note that Sample P1 and P2 are a mixture of

several natural sands. They were mixed in order to represent a “general” or “average” behavior

of several sand strata at a specific site. Samples B1 and B2 originate from a site located in the

Danube Terrace in Budapest. This location has a Miocene age base layer on which Holocene

terrace sediments were deposited in many stages. During the Holocene age, the area became a

basin shore with a flat surface full of alluvial terraces. The fluvial, coarser grained granular

materials deposited by the river were deflated and blown by the wind in short distances. How-

ever, during flood periods fluvial effects also formed the surface. Generally, samples from both


66

locations show a quite complex grain structure with heterogeneous grain shapes, see Table 4-2.

Typical grain shapes vary between sub-angular and rounded shapes.

In Table 4-1, the maximum void ratio (loosest state) was measured by pouring material

through a funnel into a mold with a diameter of 50 mm and height of 100 mm and gently lifting

the funnel to “bulk up” the soil. The minimum void ratio (densest state) was measured by filling

the mold in three stages. After each stage, the vibration platform normally used for sieving was

used to vibrate and compact the specimen while a small force was applied to the top of the

specimen with a tamping rod. At least 5 tests were done for both loose and dense states. The

maximum difference in weight was ± 5 g which would change void ratio by about 0.04. Usually

the measured values varied about ± 3 g from the average value (e ± 0.03).

Table 4-1 Soil properties of tested samples

Sam-

ple ID

Mean

part. di-

ameter

Effective

part. di-

ameter

Uni-

form.

coeff.

Fines

content

Max

void

ratio

Min

void ra-

tio

Spe-

cific

density

Liq.

limit

for

fines

Plastic

limit

for

fines

Plast.

index

for

fines

d50 [mm] d10

[mm]

CU

[-]

FC

[%]

emax

[-]

emin

[-]

ρs

[g/cm3]

wl

[%]

wp

[%]

IP

[%]

P1 0.243 0.130 2.18 5.69 0.788 0.516 2.680 -

P2 0.424 0.193 2.38 1.32 0.699 0.474 2.659 -

P3 0.365 0.160 2.74 0.25 0.677 0.453 2.660 -

P4 0.179 0.010 20.7 16.75 0.849 0.476 2.670 33.0 17.3 15.7

P5 0.322 0.151 2.50 3.26 0.762 0.488 2.664 -

P6 0.211 0.109 2.06 7.56 0.790 0.494 2.674 36.6 18.2 18.4

P7 0.107 0.013 9.85 21.11 0.949 0.524 2.708 30.4 19.7 10.7

B1 0.191 0.007 30.9 18.84 0.876 0.418 2.678 32.3 21.0 11.3

B2 0.218 0.109 2.17 3.05 0.761 0.496 2.654 -

Grain size distribution curves are shown in Figure 4-1. Most samples exhibit uniform

grain size as medium to fine sand, containing fines up to 21%. Atterberg limits of fines extracted

from five samples were measured (Table 4-1). Plasticity index ranges from 10 -18 % and sam-

ples of fines were classified as clay with low to intermediate plasticity. It is believed that similar

plasticity index is applicable for other samples where fines were not tested separately. Sample

B1 and P3 contained a significant percentage of gravel. The gravel was removed for RC testing,

but in a separate comparative study Bender Element tests were performed on both conditions

for B1: with and without the gravel (Szilvágyi, Panuska, & Ray, 2018). The typical shape of

the grains can be observed in Table 4-2.


67

Figure 4-1 Grain size curves for soils tested. (Note that P8 and B1(gr) were only tested by

bender elements in the comparative study)

Table 4-2 Photos of typical grain shapes – not to scale

P1 P2 P3

P4 P5 P6

P7 B1 B2

0

10

20

30

40

50

60

70

80

90

100

0.001 0.01 0.1 1 10

Pe

rce

nta

ge

passin

g (

%)

Particle diameter (mm)

P1

P2

P3

P4

P5

P6

P7

P8

B1

B1(gr)

B2


68

Note that the photos were taken after washing the soil, with a regular digital handheld

camera and they are not to scale; the purpose of showing them is solely to preview particle

shape of the main grains.

The testing program consisted of measurements at a wide range of void ratios and all

relevant confinements. All RC-TOSS tests were performed on dry samples under isotropic con-

finement. The testing conditions for all confinement and density states are summarized in Table

4-3.

Table 4-3 Test matrix

Void ratios listed in the table represent conditions at the beginning of a test, usually ob-

tained under 80 kPa confinement for RC. The listed confining stresses were applied as an in-

creasing sequence. One value of Dr=1.09 for Sample P2 deviates from maximum density which

could be related to slightly inaccurate volume measurements on the hollow cylinder.

RC-TOSS results were also compared to a set of independent measurements with the

bender element (BE) method performed on the same soil samples by Jakub Panuska at the De-

partment of Geotechnics at the Slovak University of Technology in Bratislava. A comprehen-

sive study was performed to compare results from the two laboratory testing methods, see

(Szilvágyi, Panuska, & Ray, 2018). Bender Element tests were performed in a triaxial cell under

comparable confinement and density conditions.

Sample ID Test type Confinement Void ratio Relative density

p' [kPa] e [-] Dr [-]

P1 RC-TOSS 77; 160 0.572 0.79 85 0.659 0.47

74; 160; 240 0.757 0.11

P2 RC-TOSS 84; 160; 220 0.454 1.09

79; 160; 240 0.663 0.16

P3 RC-TOSS 80; 160 0.637 0.14

57; 120; 180 0.596 0.34

P4 RC-TOSS 65; 73 0.617 0.62

100; 150; 200; 250; 300 0.650 0.53

P5 RC-TOSS 75; 100; 150; 200; 250; 300 0.512 0.91

78; 180 0.634 0.47

P6 RC-TOSS 75; 100; 150; 200; 250; 300 0.569 0.75

75 0.644 0.49

P7 RC-TOSS 75; 100; 150; 160; 200; 250; 300 0.571 0.89

160 0.677 0.64

B1 RC-TOSS 77; 115; 150; 180; 200; 240; 300 0.621 0.56

62; 115; 180; 240 0.761 0.25

B2 RC-TOSS 77; 100; 150; 200; 250; 300 0.512 0.94

60; 100; 150; 200; 250; 300 0.570 0.72


69

In the comparative study, piezoelectric bender elements were installed in the top cap and

bottom pedestal of a Bishop and Wesley triaxial cell. The BE could produce both shear and

compression waves, but for this study only shear waves were evaluated in detail. Three types

of excitation signals were used with peak to peak driving amplitudes of 28 V and different

frequencies. The signals were: single sine wave (3-20 kHz), four sine wave group (10-17 kHz),

and linear sine sweep burst (5-50 kHz). The different signals and frequencies were chosen to

better reveal the behavior of the entire system (triaxial device assembly, bender elements, soil

sample) (Camacho-Tauta, Cascante, Viana Da Fonseca, & Santos, 2015) and secondary effects

such as dispersion (Greening & Nash, 2004), near-field effects and overshooting (Jovicic, Coop,

& Simic, 1996). A signal stacking procedure was used for some samples with ten stacks per

one measurement to reduce noise in the signal. In BE testing data analysis is also an important

part of evaluation of results. Several methods can be employed to evaluate travel time. “Obser-

vational” methods or characteristic point methods in the time domain such as Peak-to-Peak

(PtP) and Start-to-Start (StS) are two often used approaches. Others are time domain methods

such as Cross-Correlation (CC) or frequency domain methods (FD) which use phase delay to

obtain travel time. These methods were used and compared in the evaluation process; finally

travel time values obtained by the different methods fell generally into the range of ±15%.

Further details and discussion about the comparative study can be found in (Szilvágyi, Panuska,

& Ray, 2018).

4.2 Measured Gmax compared to correlations from literature and BE

Results from all tests were examined for measurement or procedural errors before including

them in the dataset. Many preliminary tests were performed on these and other materials to

validate procedure and data analysis. These included the topics of confinement time and dry vs.

saturated testing. Figure 4-2 shows the results of an RC test lasting 20 days. The increase in

Gmax after 1000 minutes is less than 2.3% per log cycle of time which is in agreement with the

findings of (Afifi & Richart, 1973) and can be considered negligible. In all further tests a 15-

minute confinement period was applied after reaching a given confinement level, before starting

measurements.


70

Figure 4-2 Effect of confinement time on Gmax

Comparison of dry and fully saturated testing is shown in Figure 4-3. An excellent agree-

ment between obtained Gmax values and stiffness degradation can be observed.

Figure 4-3 Comparison of dry and fully saturated test

Small strain stiffness for all samples measured by RC is shown in Figure 4-4. Note that Samples

P4, P6, P7 and B1 (shown with circles) contained a considerable amount of fines (above 7%),

Gmax = 1.2586ln(t) + 117.942

R² = 0.96

0

25

50

75

100

125

150

10 100 1000 10000 100000

Gm

ax

[Mp

a]

Confinement time, t [s]

Sample B1

20

40

60

80

100

120

0.0001 0.001 0.01 0.1 1.0

Gm

ax

[MP

a]

g [%]

RC dry

TOSS dry cy1

TOSS dry cy20

RC saturated

TOSS saturated cy1

TOSS saturated cy20 Sample B1


71

while all others only a slight amount, see Table 4-1. While the soils with fines seem to show a

slightly lower Gmax value at a given confinement, this plot cannot be used to assess any trends

about fines, since the void ratios are different for almost every test. The influence of void ratio

on Gmax can be reduced by examining Gmax normalized by a void ratio function and plotting the

results versus confinement, as shown in Figure 4-5.

Figure 4-4 Small strain stiffness of all samples measured by RC. (Soils with FC>7% are

shown with circles)

The gathered correlations were compared to measurement data. The most commonly used

function to estimate Gmax is given by Equation 2-25. From that, the void ratio function given by

Equation 2-23 was used to compare the data for all soils. The obtained intrinsic parameters for

each soil are given in Table 4-4. The fits were obtained by least squares regression using MS

Excel’s Solver and the GRG approach discussed earlier. To perform the regression, measured

and estimated Gmax values first were plotted against each other using an initial estimate for A

and n. Then the squared errors were summed up and Solver changed A and n gradually until the

sum of the squared errors was minimized. As mentioned, this method works best if the fitting

parameters are the same order of magnitude, so for the fitting, n was multiplied by 100 on the

spreadsheet. The goodness of the fit was quantified by calculating the average of the relative

errors and the average of the absolute values of the relative errors in the Gmax (measured) vs.

Gmax (estimated) plot.

0

50

100

150

200

250

0 50 100 150 200 250 300 350

Gm

ax

[MP

a]

Confinement, p' [kPa]

P1

P2

P3

P4

P5

P6

P7

B1

B2


72

Table 4-4 Obtained intrinsic parameters for all tested soils (to be used with Equation 2-25)

Sample ID Mean

part.

diam.

Uni-

form.

coeff.

Fines

content

Intrinsic

parameters for

Gmax=A F(e) (p'

patm

)

n

Average

of rela-

tive er-

rors

Average of

absolute

values of

relative errors

Coefficient

of determi-

nation

d50

[mm]

CU

[-]

FC

[%]

A

[-]

n

[-]

erel,avg

[%]

|erel|avg

[%]

R2

[-]

P1 0.243 2.18 5.69 80 0.51 2.1 7.7 0.73

P2 0.424 2.38 1.32 56 0.60 -3.1 7.3 0.96

P3 0.365 2.74 0.25 56 0.51 -0.3 2.2 0.99

P4 0.179 20.7 16.75 65 0.50 0.3 4.3 0.98

P5 0.322 2.50 3.26 57 0.42 0.3 2.2 0.99

P6 0.211 2.06 7.56 61 0.41 0.5 7.1 0.91

P7 0.107 9.85 21.11 62 0.39 0.3 2.7 0.96

B1 0.191 30.86 18.84 52 0.52 1.5 8.8 0.81

B2 0.218 2.17 3.05 64 0.50 1.8 10.2 0.73

Samples

with fines* - - - 60 0.44 -0.2 8.8 0.83

Samples

without

fines**

- - - 62 0.47 0.8 12.6 0.78

All samples - - - 62 0.45 -1.1 11.2 0.71

*FC>7% **FC<7%

The fitted equations and normalized plots of Gmax can be seen in Figure 4-5 and Figure

4-6. Note that the confinement exponent n was slightly different for the fits obtained for samples

with fines, without fines and for all samples, so only one correlation can be visualized in Figure

4-6.


73

Figure 4-5 Gmax normalized by void ratio function vs. confinement for all samples.

(Soils with FC>7% are shown with circles)

Figure 4-6 Gmax normalized by confinement function vs. void ratio for all samples.

(Soils with FC>7% are shown with circles)

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300 350

Gm

ax /

F(e

) [-

]

Confinement, p' [kPa]

P1

P2

P3

P4

P5

P6

P7

B1

B2

fit for samples with FC<7%

fit for all samples

fit for samples with FC>7%

0

20

40

60

80

100

120

140

160

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

Gm

ax /

(p'/p

atm

)n[-

]

Void ratio, e' [-]

P1

P2

P3

P4

P5

P6

P7

B1

B2

fit for all samples


74

Some observations can be made based on Table 2-2, Figure 4-5 and Figure 4-6:

The obtained intrinsic parameters A and n are in the range of values found in literature for

similar soils, see Table 2-1 and Table 2-2.

The high A value for sample P1 seems out of trend. This may be because it is a mixed soil

with several grain shapes, although so is P2, for which a smaller A was obtained which is

not out of trend.

The fit for the soils with higher fines content (FC>7%, so P4, P6, P7 and B1) is not signifi-

cantly different from the fit for the soils with lower fines content or the fit for all samples,

only a slight reduction can be observed in Gmax as FC is increasing. This is due to the fact,

that fines in these soils are plastic (IP=10-18%), so the increase of FC does not result in a

decrease in Gmax as proposed by (Iwasaki & Tatsuoka, 1977) or (Wichtmann, Navarrete

Hernández, & Triantafyllidis, 2015). This tendency was also found by (Carraro, Prezzi, &

Salgado, 2009).

The fit for all samples given in Table 4-4 can be used to estimate Gmax. This is shown in

Figure 4-7, with lines of ±15%. Out of the 69 measured Gmax values 78 % lay within the

±15% margin with this correlation. 90% of the data points lay within the ±20% margin, so

the obtained correlation gives a good estimation of the measured data.


75

Figure 4-7 Measured vs. estimated Gmax using the correlation obtained in this study for all

samples in Table 4-4. Ranges of ±15% are given with dotted lines

In order to compare results to correlations for Gmax given in literature, several authors’

equations were investigated. A brief discussion follows about the most relevant ones.

The correlations obtained by (Carraro, Prezzi, & Salgado, 2009) for their sands containing

plastic fines (kaolin clay with IP=26% up to FC=10%) gave very good estimations for the soils

investigated in this study, although plasticity of fines was lower (IP=10-18%) and in three cases

FC was higher (17%, 21% and 19% for P4, P7 and B1 respectively). Results are shown in

Figure 4-8. For the samples with FC<10% the correlation coefficients for the closest FC values

in Table 2-2 were used. The fit is considered excellent, as 75 % of measured Gmax values lie

within the ±15% margin with this correlation. 87% of the data points lay within the ±20% mar-

gin.

0

50

100

150

200

0 50 100 150 200

Gm

ax

esti

mat

ed [

MP

a]

Gmax measured [MPa]


76

Figure 4-8 Measured vs. estimated Gmax using the correlations given by (Carraro, Prezzi, &

Salgado, 2009) in Table 2-2. Ranges of ±15% are given with dotted lines

A different void ratio function was suggested by (Biarez & Hicher, 1994) and while their

correlation was given for an entire soil group with wl < 50%, which suggests it was originally

obtained for fine grained soils, their correlation yields satisfactory results for the tested soils,

see Figure 4-9. Out of the 69 measured Gmax values 72 % lay within the ±15% margin with

this correlation. 84% of the data points lie within the ±20% margin. This corresponds to R2=0.65

for all tested soils.

0

50

100

150

200

0 50 100 150 200

Gm

ax

esti

mat

ed [

MP

a]

Gmax measured [MPa]


77

Figure 4-9 Measured vs. estimated Gmax using the correlation given by (Biarez & Hicher,

1994) in Table 2-3. Ranges of ±15% are given with dotted lines

The correlations given by (Wichtmann & Triantafyllidis, 2009) and (Wichtmann,

Navarrete Hernández, & Triantafyllidis, 2015) were studied in detail, since they focused on the

effects of grain size characteristics and fines content on Gmax, although the fines in their study

were nonplastic whereas the tested samples in this study have plastic fines. The authors argued

that Gmax of sands with nonplastic fines is decreasing with increasing fines content compared to

clean sands; the reduction is given by Equations 2-26 to 2-29. Otherwise their tested soils had

similar grain size properties as some samples in this study (P1, P2, P3, P5, P6, B2), so their

correlations are applicable. All other samples (P4, P7, B1) had much higher CU (from 10 to 31)

and higher FC (17; 21; 19% respectively), so they were not considered in the comparison shown

in Figure 4-10. The correlation for clean sands (Wichtmann & Triantafyllidis, 2009) tends to

over-predict many measured results. 47% of data points lie within the range of ±15% and 60%

of data points lie within the range of ±20%. The correlation given for sands with nonplastic

fines on the other hand introduces a too large reduction for these soils with a small content of

0

50

100

150

200

0 50 100 150 200

Gm

ax

esti

mat

ed [

MP

a]

Gmax measured [MPa]


78

plastic fines (up to 7.6%). Goodness of fit is similar to previous equations: 51% of data points

lay within the range of ±15% and 63% of data points lay within the range of ±20%. Equations

2-26 to 2-29 underestimate Gmax in some cases by more than 35%.

Figure 4-10 Measured vs. estimated Gmax using the correlations given by (Wichtmann &

Triantafyllidis, 2009) and (Wichtmann, Navarrete Hernández, & Triantafyllidis, 2015) in

Equations 2-26 to 2-29. Ranges of ±15% are given with dotted lines

Measured results were also compared to the correlation suggested by (Oztoprak & Bolton,

2013), see Figure 4-11. Their correlation shown in Equation 2-33 with the coefficients given in

Table 2-5 overestimated most measured results, however estimated values are within the range

of ±100% , which was also found by them for most of the gathered data.

0

50

100

150

200

0 50 100 150 200

Gm

ax

esti

mat

ed [

MP

a]

Gmax measured [MPa]

Wichtmann et al 2009

Wichtmann et al 2015

+/- 15%

+/- 35%


79

Figure 4-11 Measured vs. estimated Gmax using the correlation given by (Oztoprak & Bolton,

2013). The range of 200% and 50% is given with solid lines

Measurements have been compared to an independent set of tests performed with BE at

the Geotechnical Laboratory of the Slovak University of Technology in Bratislava by Jakub

Panuska. Since a wide variety of void ratios and confining stresses were tested, comparison and

interpretation of data were done by fitting one data set with factors accounting for void ratio

and confining stress then comparing them to the direct results of the other test. So, RC tests

were fit with parameters to allow shifts in void ratio and confining stress then compared to BE

measured results. Likewise, BE tests were fit with similar equations and compared to RC meas-

ured results. Figure 4-12 shows the confinement levels and void ratios in the comparative study.

Gm

ax

estim

ated

[M

Pa]

Gmax measured [MPa]

this study


80

Figure 4-12 Confinement and void ratio conditions in the comparative study

Equations 2-22 and 2-23 were used for regression, and obtained fitted parameters A and

n are listed in Table 4-5. Note, that in this comparison p’ is not divided by the atmospheric

pressure patm, hence the range of values of A and n are different than in the earlier correlations

shown in Table 4-4. Results obtained with the two methods are shown in Figure 4-13. All of

the measurement points for RC at different e and p´ from the testing matrix were used.

Table 4-5 Obtained intrinsic parameters by RC and BE testing for all samples

using Equation 2-22.

Sample RC BE

A n R2 A n R2

[-] [-] [-] [-] [-] [-]

P1 7.67 0.51 0.74 5.85 0.54 0.97

P2 3.51 0.60 0.96 3.81 0.55 0.99

P3 5.29 0.51 0.98 4.07 0.56 0.99

P4 6.49 0.50 0.99 3.18 0.63 0.99

P5 8.14 0.42 0.99 3.82 0.56 0.99

P6 7.36 0.45 0.97 4.34 0.58 0.98

P7 9.92 0.40 0.97 4.00 0.57 0.98

P8 - - - 5.15 0.58 0.99

B1 5.49 0.49 0.80 4.39 0.53 0.98

B1(gr) - - - 3.77 0.57 0.96

B2 6.55 0.50 0.73 5.50 0.53 0.99

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 100 200 300 400

Void

rat

io e

(-)

Confinement p' (kPa)

313 BE tests

69 RC tests


81

The agreement between measured results is generally very good. In the case of samples

with high void ratios (loose samples) RC measured higher Gmax than BE as a tendency. This

may be due to the slight inaccuracy of the measurement of void ratio in the hollow cylinder RC

samples or less complete coupling of the BE system in looser soils.

Similar comparisons between RC and BE has been made in many researches, results gath-

ered from the literature are plotted in Figure 4-14. Results from different authors fit into a close

range of ±15 % from exact agreement similarly to this study. Coordinated BE testing and com-

parison with equations proposed in (Iwasaki & Tatsuoka, 1977) based on RC testing for

Toyoura sand also showed a ±15 % difference between BE and RC tests even though there were

different evaluation procedures for BE tests.

Figure 4-13 Comparison of Gmax measured with BE and RC in this study


82

Figure 4-14 Comparison of Gmax measured with BE and RC in literature

4.3 Modulus reduction curves

Modulus reduction curves for each sample are shown in the following figures continuously. For

TOSS tests, cycle 100 is shown, except for some cases, where stated otherwise (cy10 in the

legend means cycle 10 is shown). For each soil the first figure shows the reduction curve in a

G/Gmax vs. g/gref plot. Equations 2-15 – 2-17 were used and a fit has been obtained by regression

with minimizing the sum of residual squares using MS Excel’s Solver with the GRG method

(Generalized Reduced Gradient). The fitted Ramberg-Osgood (RO) model parameters, , C, R

are shown in the legend in the top right corner together with the coefficient of determination R2

which provides a measure of how well the observed outcomes are replicated by the model. The

legend in the bottom right corner shows the identification number of each Resonant Column

(RC) and Torsional Shear (TOSS) test in brackets. The second figure shows the measurement

results in a G/Gmax vs. g plot which allows for comparison with the most commonly used refer-

ences in the literature, given by (Oztoprak & Bolton, 2013), (Vucetic & Dobry, 1991) and

(Wichtmann & Triantafyllidis, 2013). Equations 2-37 and 2-40 – 2-42 were used from the for-

mer paper to calculate a degradation curve for each soil based on grain size characteristics; this

is given with a dashed line named “Oztoprak&Bolton fit” (OB fit). It has to be noted, that this

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

Gm

ax

RC

(M

Pa)

Gmax BE (MPa)

Gu et al. (2015)

Sahaphol and Miura (2005)

Payan et al. (2016)

Ferreira et al. (2006)

Yang and Gu (2013)

Youn et al. (2008)

Cai et al. (2015)

15 %


83

fit and the curve given by (Wichtmann & Triantafyllidis, 2013) (denoted as W curve) are usu-

ally only valid for one set of measurements in the graph, since p’ is used in them and confine-

ment was not the same for all tests. Beside this the “mean” curve given in Equation 2-37 is also

shown as “Oztoprak&Bolton mean” (denoted OB mean). This curve was obtained as the best

fit for a large number of test results by the authors. The curves for soils with a PI=0%, 15% and

30% (denoted VD0, VD15, VD30) are also presented from (Vucetic & Dobry, 1991), since the

PI=0% curve suggested for sands did not fit the data well.

It should be noted, that the RO fit curve, which is obtained in the G/Gmax vs. g/gref plot, cannot

be plotted as a single curve in G/Gmax vs. g plot, because gref is different for each confinement

and void ratio. Similarly, the curves from literature cannot be compared to the RO fit directly,

because the VD curves are only given as single data points in the G/Gmax vs. g plot and gref is

unknown for them, which is also the case for the OB curves. Therefore, separate comparisons

are made to show the best fit, which is usually obtained by the RO fit and the applicability of

the curves from literature.

For Sample P1 RC and TOSS measurements gave very similar results in the range of

strains where both were applicable. Results from the separate tests also fall on a close range of

a single degradation curve in Figure 4-15. The RO fit shows a good estimation of measured

values with R2=0.94. The OB fit curve over-predicts the effect of degradation and shows ap-

prox. 20% lower values of G/Gmax than the measured ones. The OB mean curve is in quite good

agreement with measured data; estimated values are only 5-10% lower than measured ones.

The W curve is probably the best fit out of these correlations from literature, while VD15 is

also an acceptable fit.

The measured points of the degradation curve showed a larger scatter for Sample P2, see

Figure 4-17. Very low strain (g/gref<0.1) TOSS measurement results show 10-20% lower values

of Gmax than RC. This strain level is probably the limit of accuracy for the TOSS tests due to

the precision of the proximitors used. These measurement points were excluded for the RO fit

procedure and a good fit was achieved with R2=0.91. This problem does not appear with the

RC tests, since strain is measured with the accelerometer in RC tests. However, overall agree-

ment between RC and TOSS for higher strains (g/gref>0.1) is satisfactory. The OB fit curve does

not follow measured data; 35-40% lower values of G/Gmax are predicted by it than the measured

ones. Generally, the reduction of Gmax is not as pronounced as for Sample P1; while the OB

mean curve and W curve are closer to measurement trend, VD30 is the best fit for Sample P2.


84

A similar trend can be observed for Sample P3 as described for P2. Three very low strain

measurement points were excluded here as well, but otherwise the RO fit is excellent, with an

R2=0.97, see Figure 4-19. Once again, the scatter in the G/Gmax vs. g plot in Figure 4-20 is

slightly higher. The OB fit curve shows a too strong reduction, OB mean and W curves are

closer to the measured data, while the bigger scatter means that measured points lay between

the VD0 and VD30 curves, with the VD15 being a good fit for them.

Measured points for Sample P4 show the largest scatter, with an R2=0.89 fit of the RO

model. Agreement between RC and TOSS is acceptable; see Figure 4-21, although not as good

as for the other samples. The OB fit coincides with VD0 curve which are both a lower bound

of measured data, while upper bound would be the VD30 curve, with VD15 being a quite good

fit and the OB mean the best fit for Sample P4 out of these curves from literature. The W curve

shows a very significant reduction, which is not following measured trends. This correlation

takes the effect of uniformity coefficient on G/Gmax into account, but for samples with a CU >3

(P4, P7 and B1) the effect is significantly over predicted.

For Samples P5 and P6 a lower number of measured data points were available, because

the testing equipment malfunctioned during TOSS tests No. (24) and (25). Unfortunately, this

became only clear after performing the test, during post-processing of data. Meanwhile RC

measurements are shown from these tests, since they were expected not to be affected by the

error. Degradation curve obtained for P5 is fitted with a RO curve with R2=0.93, see Figure

4-23. Due to the lower number of data points at higher strains, correlations from the literature

are difficult to assess; but once again a curve between VD15 and VD30 would be a fair estimate

and W curve is also acceptable. For P6 the RO fit is very good, with R2=0.97, see Figure 4-25.

From the literature here again, OB mean and W curve could be considered to be the best fits

for P6.

The degradation curve for Sample P7 is shown in Figure 4-27 and Figure 4-28. Agree-

ment between RC and TOSS for a single test is excellent, while the scatter between tests with

different confinements is somewhat larger. The normalization used for Figure 4-27 brings the

measured points only slightly closer this time and a RO fit could be found with R2=0.93. Some

measured points lay even above the VD30 curve and while the OB mean and OB fit curves lay

on each other, none of them are a good fit, a curve between VD15 and VD30 would be the best

fit. The W curve once again shows a more significant degradation which does not follow meas-

ured behavior, note that CU was higher than 3 for this sample.

Results for Sample B1 again show the benefits of using the normalized plots; see Figure

4-29 and Figure 4-30. Once again, the RO fit is exceptionally good with R2=0.98 if the very


85

low strain TOSS measurements are excluded. The trend for the curves are similar as before:

OB fit show a too significant degradation, OB mean can be considered a lower bound for meas-

ured data, while most of the data are between the VD15 and VD30 curves. The W curve is again

far from measured data as with the previous sample.

Similar general observations can be made about Sample B2, except the W curve is much

closer to measured values. The RO fit is very good with R2=0.96 and most data points are be-

tween the VD15 and VD30 curves while other curves show a too significant degradation.


86

Figure 4-15 Modulus reduction curve for Sample P1 with RO fit

Figure 4-16 Modulus reduction compared to correlations for Sample P1

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(31)RC

(31)TOSS

(32)RC

(32)TOSS

(33)RC

(33)TOSS

RO fit

Sample P1

=0.9160

C=0.7715

R=2.155

Coeff. of

determ.

R2=0.94

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g[%]

(31)RC

(31)TOSS

(32)RC

(32)TOSS

(33)RC

(33)TOSS

Oztoprak&Bolton fit

Oztoprak&Bolton mean

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample P1


87



0.0

0.2

0.4

0.6

0.8

1.0

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(34)RC

(34)TOSS

(35)RC

(35)TOSS

RO fit

Sample P2

=0.7798

C=0.8622

R=2.113

Coeff. of

determ.

R2=0.91

excluded from fit

0.0

0.2

0.4

0.6

0.8

1.0

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g[%]

(34)RC

(34)TOSS

(35)RC

(35)TOSS

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample P2


88



0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(10)RC

(10)TOSS

(11)RC

(11)TOSS

(16)RC

(16)TOSS

RO fit

excluded from fit

Sample P3

=1.0313

C=0.6029

R=2.136

Coeff. of

determ.

R2=0.97

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g[%]

(10)RC

(10)TOSS

(11)RC

(11)TOSS

(16)RC

(16)TOSS

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample P3


89



0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

ma

x[-

]

g/gref [-]

(01)RC

(01)TOSS cy10

(02)RC

(14)RC

(14)TOSS

(23)RC

(23)TOSS

RO fit

Sample P4

=1.1640

C=0.5169

R=2.244

Coeff. of

determ.

R2=0.87

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g[%]

(01)RC

(01)TOSS cy10

(02)RC

(14)RC

(14)TOSS

(23)RC

(23)TOSS

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample P4


90



0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(15)RC

(15)TOSS

(24)RC

RO fit

Sample P5

=1.1553

C=0.6274

R=2.232

Coeff. of

determ.

R2=0.93

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1.0

G/G

max

[-]

g[%]

(15)RC

(15)TOSS

(24)RC

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample P5


91



0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(17)RC

(17)TOSS

(25)RC

RO fit

Sample P6

=1.1323

C=0.5357

R=2.431

Coeff. of

determ.

R2=0.97

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g[%]

(17)RC

(17)TOSS

(25)RC

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample P6


92



0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(18)RC

(18)TOSS

(26)RC

(26)TOSS

RO fit

Sample P7

=1.1221

C=0.6755

R=2.049

Coeff. of

determ.

R2=0.93

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g[%]

(18)RC

(18)TOSS

(26)RC

(26)TOSS

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample P7


93

Figure 4-29 Modulus reduction curve for Sample B1 with RO fit

Figure 4-30 Modulus reduction compared to correlations for Sample B1

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(27)RC

(27)TOSS

(30)RC

(30)TOSS

RO fit

excluded from fit

Sample B1

=1.0757

C=0.5655

R=2.454

Coeff. of

determ.

R2=0.98

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g [%]

(27)RC

(27)TOSS

(30)RC

(30)TOSS

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample B1

excluded from fit


94

Figure 4-31 Modulus reduction curve for Sample B2 with RO fit

Figure 4-32 Modulus reduction compared to correlations for Sample B2

0.0

0.2

0.4

0.6

0.8

1.0

0.001 0.01 0.1 1.0 10.0

G/G

max

[-]

g/gref [-]

(28)RC

(28)TOSS

(29)RC

(29)TOSS cy20

RO fit

Sample B2

=0.8186

C=0.9036

R=2.211

Coeff. of

determ.

R2=0.96

0.0

0.2

0.4

0.6

0.8

1.0

0.0001 0.001 0.01 0.1 1.0

G/G

ma

x[-

]

g[%]

(28)RC

(28)TOSS

(29)RC

(29)TOSS cy20

Oztoprak&Bolton fit


Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Wichtmann 2013

Sample B2


95

All 279 measured points of G/Gmax for all soils are shown in a single plot in Figure 4-33

with a RO fit for all points. The scatter of the points is significant, but a similar range was found

in many tests shown in (Oztoprak & Bolton, 2013), see Figure 2-20. At a given strain the scatter

is around ±15% in G/Gmax value. However, the single RO fit for all soils is satisfactory with

R2=0.89.

Figure 4-33 Modulus reduction curve for all samples with RO fit

The parameters of the obtained RO fits and the coefficient of determination are summa-

rized in Table 4-4 and all curves are plotted together in Figure 4-34.

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

ma

x[-

]

g/gref [-]

all samples

RO fit for all samples

All samples

=0.8332

C=0.6412

R=2.127

Coeff. of

determ.

R2=0.89


96

Table 4-6 Obtained Ramberg-Osgood material model parameters for all tested soils

Sample ID Mean par-

ticle diam-

eter

Uniformity

coefficient

Fines con-

tent

RO parameters Coefficient

of determi-

nation

d50

[mm]

CU

[-]

FC

[%] [-]

C

[-]

R

[-]

R2

[-]

P1 0.243 2.18 5.69 0.9160 0.7715 2.155 0.94

P2 0.424 2.38 1.32 0.7798 0.8622 2.113 0.91

P3 0.365 2.74 0.25 1.0313 0.6029 2.136 0.97

P4 0.179 20.7 16.75 1.1640 0.5169 2.244 0.87

P5 0.322 2.50 3.26 1.1553 0.6274 2.232 0.93

P6 0.211 2.06 7.56 1.1323 0.5357 2.431 0.97

P7 0.107 9.85 21.11 1.1221 0.6755 2.049 0.93

B1 0.191 30.86 18.84 1.0757 0.5655 2.454 0.98

B2 0.218 2.17 3.05 0.8186 0.9036 2.211 0.96

Fit for all

soils - - - 0.8334 0.6386 2.142 0.89

Figure 4-34 RO curves for each soil and the fit for all soils

No significant dependence of the degradation curve on grain size characteristics can be

identified from Figure 4-34, which is perhaps because the curves are fairly close to each other

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1.0 10.0

G/G

ma

x[-

]

g/gref [-]

P1

P2

P3

P4

P5

P6

P7

B1

B2



97

and the scatter in the measurements for a single soil could result in this slight scatter of the

degradation curves. This narrow band of degradation curves suggests that the fit obtained for

all samples can be considered to be applicable to typical Danube sands.

4.4 Damping curves

The following figures show measured damping values for each soil. Damping was measured in

RC testing with the log decrement method and in TOSS testing as hysteretic damping as dis-

cussed earlier. It should be noted, that damping values from TOSS tests shown here are calcu-

lated from the hysteresis loop of cycle 100. It has to be noted also, that there is a difference

between the RC damping obtained by the log decrement method and hysteretic damping ob-

tained by TOSS. While hysteresis loops obtained by TOSS are symmetrical, strain level is clear,

in RC the free decay results in a continuously collapsing hysteresis, and hence the strain level

is continuously decreasing. For TOSS testing the effect of cycle number has to be considered.

If the response in TOSS test changes during cycles, it is suggested to take a hysteresis loop for

assessment, where the response corresponds to a “steady state” and not the first couple of cy-

cles. These two effects result in an inherent uncertainty when comparing results of both methods

adding to the difficulty of measurement. The scatter in damping is therefore usually bigger than

in the modulus degradation. This is found in literature too. While the uncertainty of measure-

ments is not reassuring, it has to be noted, that the effect of damping on practical calculations

such as ground response analysis is minor compared to modulus reduction and a scatter of ±2%

in the value of D can be considered typical.

Two figures are shown for each soil continuously; the first shows damping vs. strain com-

pared to damping curves given in (Vucetic & Dobry, 1991) (denoted as VD0, VD15 and VD30).

The second figure shows measured damping values compared to estimated values by the RO

fit obtained based on the degradation curves, given in Table 4-6; as well as estimated values

based on the correlations of (Ishibashi & Zhang, 1993), (Zhang, Andrus, & Juang, 2005) and

(Wichtmann & Triantafyllidis, 2013). In this figure, ±3% in the value of D is shown with dotted

lines.

Altogether 261 values of damping have been measured for all soils either with RC or with

TOSS. Some general tendencies can be observed from Figure 4-35 to Figure 4-52. Measured

damping values are generally lower than the predictions of the VD0 curve. Even the VD15

curve seems to be an upper bound to measured values. The damping values predicted by the

RO fits above D = 2% are approx. 5% higher, than the measured values and a similar trend is


98

true for the correlation of (Ishibashi & Zhang, 1993). The correlations of (Zhang, Andrus, &

Juang, 2005) and (Wichtmann & Triantafyllidis, 2013) provide better estimates of measured

values and in the most cases are within ±3% in the value of D. However for sample B1, which

has the highest CU value, Wichtmann’s correlation considerably underestimates damping. Gen-

erally, out of the correlations from literature, the one provided by (Zhang, Andrus, & Juang,

2005) describes damping behavior of Danube sands the best.


99

Figure 4-35 Damping curve for Sample P1 compared to correlations

Figure 4-36 Measured vs. estimated values for damping with RO fit for Sample P1

0

2

4

6

8

10

12

14

0.0001 0.001 0.01 0.1 1.0

Dam

pin

gD

[%

]

g[%]

(31)RC

(31)TOSS

(32)RC

(32)TOSS

(33)RC

(33)TOSS

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample P1

0

4

8

12

16

0 4 8 12 16

Des

tim

ated

[%

]

D measured [%]

P1 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample P1

=0.9160

C=0.7715

R=2.155


100



0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%]

g[%]

(34)RC

(34)TOSS

(35)RC

(35)TOSS

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample P2

0

4

8

12

16

0 4 8 12 16

Des

tim

ated

[%

]

D measured [%]

P2 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample P2

=0.7798

C=0.8622

R=2.113


101



0

2

4

6

8

10

12

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g[%]

(10)RC

(10)TOSS

(11)RC

(11)TOSS

(16)RC

(16)TOSS

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample P3

0

4

8

12

16

0 4 8 12 16

Des

tim

ated

[%

]

D measured [%]

P3 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample P3

=1.0313

C=0.6029

R=2.136


102



0

2

4

6

8

10

12

14

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g[%]

(01)RC

(02)RC

(14)RC

(14)TOSS

(23)RC

(23)TOSS

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample P4

0

4

8

12

16

0 4 8 12 16

Des

tim

ated

[%

]

D measured [%]

P4 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample P4

=1.1640

C=0.5169

R=2.244


103



0

2

4

6

8

10

12

14

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g[%]

(15)RC

(15)TOSS

(24)RC

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample P5

0

4

8

12

16

0 4 8 12 16

Des

tim

ated

[%

]

D measured [%]

P5 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample P5

=1.1553

C=0.6274

R=2.232


104



0

2

4

6

8

10

12

14

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g[%]

(17)RC

(17)TOSS

(25)RC

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample P6

0

4

8

12

16

0 4 8 12 16

Des

tim

ated

[%

]

D measured [%]

P6 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample P6

=1.1323

C=0.5357

R=2.431


105



0

2

4

6

8

10

12

14

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g[%]

(18)RC

(18)TOSS

(26)RC

(26)TOSS

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample P7

0

5

10

15

20

25

0 5 10 15 20 25

Des

tim

ated

[%

]

D measured [%]

P7 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample P7

=1.1221

C=0.6755

R=2.049


106

Figure 4-49 Damping curve for Sample B1 compared to correlations

Figure 4-50 Measured vs. estimated values for damping with RO fit for Sample B1

0

2

4

6

8

10

12

14

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g [%]

(27)RC

(27)TOSS

(30)RC

(30)TOSS

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample B1

0

5

10

15

20

25

0 5 10 15 20 25

Des

tim

ated

[%

]

D measured [%]

B1 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample B1

=1.0757

C=0.5655

R=2.454


107

Figure 4-51 Damping curve for Sample B2 compared to correlations

Figure 4-52 Measured vs. estimated values for damping with RO fit for Sample B2

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g[%]

(28)TOSS

(29)RC

(29)TOSS cy20

Vucetic&Dobry PI=0

Vucetic&Dobry PI=15

Vucetic&Dobry PI=30

Sample B2

0

4

8

12

16

0 4 8 12 16

Des

tim

ated

[%

]

D measured [%]

B2 RO fit

Ishibashi 1993

Zhang 2005

Wichtmann 2013

Sample B2

=0.8186

C=0.9036

R=2.211


108

Figure 4-53 Measured damping values from all samples compared to correlations

Figure 4-54 Measured damping values from all samples with RO fit

0

5

10

15

20

25

0.0001 0.001 0.01 0.1 1.0

Dam

pin

g D

[%

]

g [%]

all samples

Vucetic&Dobry PI=0

Vucetic&Dobry PI=30

0.0

5.0

10.0

15.0

20.0

0.001 0.01 0.1 1.0 10.0

Dam

pin

g D

[%

]

g/gref [-]

all samples



109

Figure 4-55 shows estimated D values vs. measured values. In this plot the closer the

values are to the 45° line, the better the prediction. Coefficients of determination R2 can also be

calculated based on this plot. Obtained values of R2 were 0.15 for the predictions of RO and

(Ishibashi & Zhang, 1993), 0.78 for (Zhang, Andrus, & Juang, 2005) and 0.57 for (Wichtmann

& Triantafyllidis, 2013). The correlation by (Zhang, Andrus, & Juang, 2005) can be considered

acceptable, however for practical use, some modifications could be beneficial. The correlation

given by Equation 2-48 was obtained by fitting the parameters c1 and c2 to many RC and TOSS

test results and separate values were suggested to be used for RC and TOSS tests.

Figure 4-55 Measured vs. estimated values for damping with RO fit for each soil tested in this

study

For practical calculations, a single set of constants would be more beneficial, therefore a

regression analysis was performed in MS Excel with the GRG method and the following equa-

tion was obtained based on the measurements in this study:

D [-] = 0.1667 (G

Gmax)

2

– 0.3932 (G

Gmax) – 0.2165 Equation 4-1

This equation was obtained in the same form as (Zhang, Andrus, & Juang, 2005) proposed

Equation 2-48, with the constants c1 = 0.1667, c2 = –0.3932 and Dmin = 0.01. Measured and

estimated values with Equation 5-4 for all tested soils are shown in Figure 4-56. Coefficient of

0

5

10

15

20

0 5 10 15 20

D e

stim

ated

[%

]

D measured [%]

RO

Zhang 2005

Wichtmann 2013

+/- 3%


110

determination R2 based on this plot is 0.82, so the estimation is slightly better than the correla-

tion of (Zhang, Andrus, & Juang, 2005) and hence it is suggested to be used for typical Danube

sands.

Figure 4-56 Measured vs. estimated values of damping with Equation 5-4

5 Thesis statements

Thesis statements were obtained based on the state-of-the-art combined Resonant Column and

Torsional Shear Testing of nine air dry (wmax<2%), typical Danube Sands containing a small

amount ((Cl+Si)<21 %) of fine particles with various plasticity (10%<IP<18 %), and compris-

ing of sub-angular and rounded grains with 0,11<d50<0,42 mm. 69 RC tests and 19 TOSS tests

were performed on samples with wide range of densities (0,11 < ID < 0,95) under relevant con-

finement pressures (80kPa<p’<300 kPa). This resulted in 69 measured values of Gmax, 279

measured values of G/Gmax and 261 measured values of D.

0

5

10

15

20

0 5 10 15 20

D e

stim

ated

[%

]

D measured [%]

all samples

+/- 3%


111

5.1 Thesis #1

I have shown that the further developed, combined Resonant Column-Torsional Shear

Device with the testing procedure presented in this dissertation, which includes device

assembly, preparation of hollow cylindrical sample, testing sequence and measurement

result interpretation; is capable of measuring the very small strain shear stiffness Gmax;

the modulus reduction curve and the damping curve of dry sands.

I have compared the obtained very small strain shear stiffness values to an independent

set of measurements performed in a separate laboratory with the Bender Element Method by a

separate researcher; and obtained acceptable agreement with 78% of the measured data within

the ±15% deviation and 91% of the measured data within the ±20% deviation.

References: (Szilvágyi Z. , 2012), (Ray, R. P., Szilvágyi, Z., 2013), (Szilvágyi, Z., Hudacsek

P., Ray, R.P, 2016), (Szilvágyi, Panuska, & Ray, 2018)

5.2 Thesis #2

I have developed a procedure to obtain the model parameters for the Ramberg-Osgood material

model using small strain shear stiffness, modulus reduction and damping properties of a Danube

Sand measured by Resonant Column and Torsional Shear Testing methods, by solving the in-

verse problem using MS Excel Solver’s Generalized Reduced Gradient Method for minimizing

the sum of squared errors between predicted and measured values of a hysteresis loop.

I have created a three-dimensional finite element model of the Torsional Shear Test

and verified that the Ramberg-Osgood material model implemented in the software

Midas GTS NX v2014 is capable of modeling the static Torsional Shear Test performed

on a hollow cylinder sample with representation of the nonlinear material behavior con-

cerning modulus reduction with increasing strain.

The verification by the back analysis of the Torsional Shear Test has not been done before

with a nonlinear material model and indicates that the Ramberg-Osgood material model can be

used in practical calculations requiring the modeling of stiffness degradation.

References: (Szilvágyi Z. , 2010), (Szilvágyi & Ray, 2018)


112

5.3 Thesis #3

I have compared the very small strain stiffness (Gmax) of typical Danube sands at a wide range

of densities and relevant confinements obtained by the Resonant Column method to correlations

from literature and confirmed that out of the presented existing correlations the correlation

given by (Carraro, Prezzi, & Salgado, 2009) gives the best estimate for Gmax.

I have confirmed based on my measurement results with the Resonant Column method,

that correlations which have been obtained for sands with non-plastic fines (Iwasaki &

Tatsuoka, 1977) and (Wichtmann, Navarrete Hernández, & Triantafyllidis, 2015), could under-

estimate Gmax of sands with plastic fines by 30-35%.

This suggests, that over the content of fine particles, the plasticity of them has an even

more dominant effect on Gmax.

I have measured the very small strain stiffness (Gmax) of typical Danube sands at a

wide range of densities and relevant confinements with the Resonant Column method and

obtained the following correlation for Gmax:

Gmax [MPa] = 62 (2.17 – e)

2

1 + e(

p'

patm)

0.45

Equation 5-1

where e is void ratio, p’ is the mean effective stress in [kPa], and patm is the atmospheric pressure

in [kPa]. The applicability of the most commonly used void ratio function in literature was

confirmed by my measurement results and was used in the above equation. Coefficient of de-

termination is R2=0.71.

This correlation can be used in practical calculations for the estimation of Gmax of typical

Danube sands, in the absence of measurements.



5.4 Thesis #4

I have measured the stiffness degradation (G/Gmax) of typical Danube sands at a wide

range of densities and relevant confinements with the combined Resonant Column and

Torsional Shear test methods and obtained the following degradation curve formulated

in the Ramberg-Osgood material model:


113

G

Gmax =

1

1+0.8334|τ

0.6386 τmax|1.142 Equation 5-2

where G is the secant shear modulus, Gmax is the very small strain stiffness, t is shear stress,

tmax is shear strength. Coefficient of determination R2 is 0.89.

This degradation curve can be used in practical calculations to describe the stiffness deg-

radation behavior of typical Danube sands.



5.5 Thesis #5

I have measured the damping (D) increase with strain of typical Danube sands at a wide

range of densities and relevant confinements with the Resonant Column and Torsional

Shear test method and obtained the following correlation for damping:

D [-] = 0.1667 (G

Gmax)

2

– 0.3932 (G

Gmax) – 0.2165 Equation 5-3

where G/Gmax is the stiffness degradation value at a given strain level. Coefficient of determi-

nation is R2=0.82.

This correlation can be used in practical calculations for the estimation of D of typical

Danube sands, in the absence of measurements.



6 Summary and future research

In my dissertation I have presented, how small strain stiffness of Danube sands can be described

and modeled based on state-of-the art laboratory measurements. A detailed literature review on

the modeling of small strain stiffness of soils has been presented to provide insight into the most

commonly used model parameters; and their determination based on correlations has also been

investigated. A comprehensive laboratory testing program has been performed on nine selected

typical Danube Sands with the combined Resonant Column-Torsional Shear device. Measure-

ment results have been compared to an independent set of measurements performed by a fellow


114

PhD student at a different laboratory with the Bender Element test method. Very good agree-

ment has been found between the results. Besides testing, data interpretation has also been de-

veloped by solving the inverse problem of choosing most suitable model parameters, which

provide the closest behavior with the used material model as the measurements. For this, the

3D finite element model of the Torsional Shear Test has been developed and the modeling of a

TOSS test has been performed with use of the nonlinear Ramberg-Osgood material model. Fi-

nally, measurement results have been presented and new correlations have been obtained for

the most important parameters, which can be used to estimate them in the absence of measure-

ments.

It has been emphasized, that some features of this complex behavior still need investiga-

tions. Some of these topics can be further investigated in laboratory; some can be modeled with

the presented FEM modeling process. Laboratory testing should focus on the effects of irregular

loading, especially RC testing after TOSS prestraining and perhaps a more precise measurement

method of void ratio could be developed. Comparison of laboratory measurements to in-situ

testing is another difficult task in this topic and needs further research. Effects of uneven dis-

tribution of density, imperfections due to sample preparation could be investigated by further

modeling.

An additional topic concerns the application of dynamic soil parameters in modeling,

especially in ground response analysis, which can be used to assess the specific value of the

earthquake load, based on local soil conditions. Our first studies in this field (Kegyes-Brassai,

Wolf, Szilvágyi, & Ray, 2017), (Szilvágyi, et al., 2017) confirmed, that the nonlinear behavior

of surface near soil layers have a major effect on earthquake loading and if the profile is not

homogeneous or nearly so, a substantial degree of amplification can be expected.


115

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List of figures

Figure 2-1 Ratio of dynamic and static moduli presented by (Alpan, 1970) ............................. 6

Figure 2-2 External and local measurements in a triaxial test: setup (left) and deviatoric stress-

strain curves with zoom into small strains (right) (Puzrin, 2012) ......................................... 6

Figure 2-3 Stiffness degradation of soil with typical strain ranges after ................................... 8

Figure 2-4 Seismic waves (original image by Prof. L. Braile ©) .............................................. 8

Figure 2-5 Propagation of shear waves in layered medium and induced deformation .............. 9

Figure 2-6 Deviatoric stress-strain curve: true (left) and normalized (right) (Puzrin, 2012) ... 11

Figure 2-7 Nonlinear stress-strain curve for initial loading and hysteresis curve for cyclic

loading ................................................................................................................................. 12

Figure 2-8 Secant shear modulus and tangent shear modulus ................................................. 14

Figure 2-9 Hysteresis loop of one cycle showing dissipated energy WD, and equivalent elastic

strain energy WS .................................................................................................................. 15

Figure 2-10 Hysteresis loops at different strain levels due to harmonic cyclic loading .......... 16

Figure 2-11 Hysteresis loops in symmetric (a) and irregular loading (b) according to the

extended Masing rules (redrawn from (Benz, 2006), after (Pyke, 1979)) ........................... 18

Figure 2-12 Stress-strain curves obtained with the Ramberg-Osgood model .......................... 19

Figure 2-13 Normalized secant and tangent modulus vs. strain obtained with the Ramberg-

Osgood model ...................................................................................................................... 20

Figure 2-14 Mohr diagram for maximum shear stress ............................................................. 21

Figure 2-15 Reduction factor B to account for higher CU and FC obtained ............................ 30

Figure 2-16 Effect of plasticity on the small strain shear modulus of mixtures of Ottawa sand

with fines based on (Carraro, Prezzi, & Salgado, 2009). Note: Clayey sand is sand with

plastic fines; silty sand is sand with nonplastic fines .......................................................... 33

Figure 2-17 Modulus reduction curves for sand ...................................................................... 37

Figure 2-18 Modulus degradation curves for clays and sand (PI=0) ....................................... 37

Figure 2-19 Effect of cyclic stiffness degradation on modulus reduction curves of clays and

sand (PI=0) by (Vucetic & Dobry, 1991) ............................................................................ 38

Figure 2-20 Best fit modulus degradation curves for sands by (Oztoprak & Bolton, 2013) ... 40

Figure 2-21 Damping curves for sand by (Seed & Idriss, 1970) and (Vucetic & Dobry, 1991)

............................................................................................................................................. 43

Figure 2-22 Damping curves for clays and sand by (Vucetic & Dobry, 1991) ....................... 44

Figure 3-1 Combined Resonant Column – Torsional Shear Device (RC-TOSS) .................... 47


125

Figure 3-2 Cross section of RC-TOSS testing device .............................................................. 48

Figure 3-3 Raw RC testing data in the measurement controlling spreadsheet ......................... 52

Figure 3-4 Interpreted TOSS measurement data. Two confinement stages and several shear

stress amplitudes are shown with cycle 1 and cycle 100 ..................................................... 52

Figure 3-5 Effect of number of cycles in TOSS test ................................................................ 53

Figure 3-6 Stress-strain response to an irregular load history in TOSS test ............................ 54

Figure 3-7 Numerical integration for obtaining hysteretic damping with the Ramberg-Osgood

material model ..................................................................................................................... 54

Figure 3-8 Modulus reduction curve obtained with Resonant Column-Torsional Simple Shear

device ................................................................................................................................... 56

Figure 3-9 Hysteresis loops of Torsional Shear tests at different stress (strain) levels............ 57

Figure 3-10 Detail of finite element mesh of soil sample and steel ring (partially invisible) .. 57

Figure 3-11 Distribution of horizontal stresses after confinement activation .......................... 60

Figure 3-12 Comparison of shear stresses in cross section ...................................................... 60

Figure 3-13 Stress-strain curve by FEM calculation and MIDAS Ramberg-Osgood .............. 61

Figure 3-14 Deformed mesh at turnaround point of loading.................................................... 61

Figure 3-15 Distribution of total displacements at turnaround point of loading ...................... 62

Figure 3-16 Shear strains at turnaround point of loading ......................................................... 62

Figure 3-17 Manual load stepping used for stable calculation of hysteresis loop ................... 63

Figure 3-18 Hysteresis loop obtained with FEM calculation ................................................... 63

Figure 3-19 Hysteresis loops obtained from two-way sequential loading in Midas ................ 64

Figure 3-20 Modulus reduction curve by FEM calculation and RC-TOSS tests ..................... 64

Figure 4-1 Grain size curves for soils tested. (Note that P8 and B1(gr) were only tested by

bender elements in the comparative study) ......................................................................... 67

Figure 4-2 Effect of confinement time on Gmax ....................................................................... 70

Figure 4-3 Comparison of dry and fully saturated test ............................................................. 70

Figure 4-4 Small strain stiffness of all samples measured by RC. (Soils with FC>7% are

shown with circles) .............................................................................................................. 71

Figure 4-5 Gmax normalized by void ratio function vs. confinement for all samples. .............. 73

Figure 4-6 Gmax normalized by confinement function vs. void ratio for all samples. .............. 73

Figure 4-7 Measured vs. estimated Gmax using the correlation obtained in this study for all

samples in Table 4-4. Ranges of ±15% are given with dotted lines .................................... 75

Figure 4-8 Measured vs. estimated Gmax using the correlations given by (Carraro, Prezzi, &

Salgado, 2009) in Table 2-2. Ranges of ±15% are given with dotted lines ........................ 76


126

Figure 4-9 Measured vs. estimated Gmax using the correlation given by (Biarez & Hicher,

1994) in Table 2-3. Ranges of ±15% are given with dotted lines ....................................... 77

Figure 4-10 Measured vs. estimated Gmax using the correlations given by (Wichtmann &

Triantafyllidis, 2009) and (Wichtmann, Navarrete Hernández, & Triantafyllidis, 2015) in

Equations 2-26 to 2-29. Ranges of ±15% are given with dotted lines ................................ 78

Figure 4-11 Measured vs. estimated Gmax using the correlation given by (Oztoprak & Bolton,

2013). Ranges of 200% and 50% are given with solid lines ............................................... 79

Figure 4-12 Confinement and void ratio conditions in the comparative study ........................ 80

Figure 4-13 Comparison of Gmax measured with BE and RC in this study .............................. 81

Figure 4-14 Comparison of Gmax measured with BE and RC in literature ............................... 82

Figure 4-15 Modulus reduction curve for Sample P1 with RO fit ........................................... 86

Figure 4-16 Modulus reduction compared to correlations for Sample P1 ............................... 86













Figure 4-29 Modulus reduction curve for Sample B1 with RO fit .......................................... 93

Figure 4-30 Modulus reduction compared to correlations for Sample B1 ............................... 93

Figure 4-31 Modulus reduction curve for Sample B2 with RO fit .......................................... 94

Figure 4-32 Modulus reduction compared to correlations for Sample B2 ............................... 94

Figure 4-33 Modulus reduction curve for all samples with RO fit .......................................... 95

Figure 4-34 RO curves for each soil and the fit for all soils .................................................... 96

Figure 4-35 Damping curve for Sample P1 compared to correlations ..................................... 99

Figure 4-36 Measured vs. estimated values for damping with RO fit for Sample P1.............. 99

Figure 4-37 Damping curve for Sample P2 compared to correlations ................................... 100

Figure 4-38 Measured vs. estimated values for damping with RO fit for Sample P2............ 100


127











Figure 4-49 Damping curve for Sample B1 compared to correlations .................................. 106

Figure 4-50 Measured vs. estimated values for damping with RO fit for Sample B1 ........... 106

Figure 4-51 Damping curve for Sample B2 compared to correlations .................................. 107

Figure 4-52 Measured vs. estimated values for damping with RO fit for Sample B2 ........... 107

Figure 4-53 Measured damping values from all samples compared to correlations .............. 108

Figure 4-54 Measured damping values from all samples with RO fit ................................... 108

Figure 4-55 Measured vs. estimated values for damping with RO fit for each soil tested in this

study................................................................................................................................... 109

Figure 4-56 Measured vs. estimated values of damping with Equation 5-4 .......................... 110


128

List of symbols

a Curve fitting coefficient

apeak Peak acceleration

A Curve fitting coefficient

Aloop Area of hysteresis loop

b Parameter used to define general Ramberg-Osgood relationship

B Reduction factor of Gmax to account for fines content

c' Cohesion

c1 Coefficient used in Damping correlations

c2 Coefficient used in Damping correlations

C Ramberg-Osgood model parameter

Cl Mass percentage of clay particles in sample

CU Uniformity coefficient

dpeak Peak displacement

d10 Effective particle diameter

d50 Mean or average particle diameter

d60 Particle diameter at which 60% of a sample's mass is comprised of smaller

particles

D Damping ratio

Di Inner diameter of sample

Do Outer diameter of sample

Dr Relative density

e Void ratio

emax Maximum void ratio

emin Minimum void ratio

erel,avg Average of relative errors

|erel|,avg Average of absolute values of relative errors

Ed Dynamic modulus

Es Static modulus of elasticity

f1 Function used for numerical integral of damping

f2 Function used for numerical integral of damping

F(e) Void ratio function


129

FC Fines content

G Shear modulus

G0 or Gmax Maximum shear modulus or very small strain shear modulus

Gsec Secant shear modulus

Gtan Tangent shear modulus

hmax Ramberg-Osgood model parameter in Midas formulation

I Mass polar moment of inertia of sample

Ip Plasticity index of fine particles in sample

I0 Mass polar moment of inertia of device

ID Density index

k Curve fitting coefficient

K0 Jáky’s at rest earth pressure coefficient

K2,max Coefficient for Gmax estimation

L Height of sample

n Curve fitting coefficient

N Number of load cycles

NG Material constant accounting for cyclic increase of Gmax

OCR Over-consolidation ratio

p' Mean effective stress

pa or patm Atmospheric pressure

PI Plasticity index

q Deviatoric stress

qL Deviatoric limiting stress

qy Deviatoric yield stress

R Ramberg-Osgood model parameter

R2 Coefficient of determination

Si Mass percentage of silt particles in sample

vp P-wave velocity

vs Shear wave velocity

wl Liquid limit

wmax Maximum water content

wp Plastic limit

WD Dissipated energy


130

WS Maximum strain energy of equivalent elastic material

x Function used to define Ramberg-Osgood relationship

y Argument of function used to define Ramberg-Osgood relationship

zN Acceleration amplitude at cycle N

Z Acceleration amplitude reading

Ramberg-Osgood model parameter

Ramberg-Osgood model parameter in Midas formulation/device constant

Logarithmic decrement

1 Major principal strain

Minor principal strain

s Shear strain (in triaxial stress state)

sr Relative shear strain (in triaxial stress state)

sL Limiting shear strain (in triaxial stress state)

’ Friction angle

g or gs Shear strain

gc Cyclic shear strain

gd Dry density

ge Elastic threshold strain

gp or g1 Shear strain at turnaround point

gr or gref Reference shear strain

gref,1 Reference shear strain at 100 kPa confinement

gstatic Static shear strain

Poisson’s ratio

Density

s Specific density

'0 Average confinement pressure

1 Major principal normal stress

3 Minor principal normal stress

v Effective vertical overburden pressure

t Shear stress


131

tmax Shear strength

tp or t1 Shear stress at turnaround point

tstatic Static shear stress

n Resonant frequency

Documents

Zsolt Szilvágyi - Széchenyi István University · Zsolt Szilvágyi Dynamic Soil Properties of Danube Sands ii Acknowledgments First, I would like to extend my appreciation and gratitude